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Substitution Method

The substitution method is most useful for systems of 2 equations in 2 unknowns. The main idea here is that we solve one of the equations for one of the unknowns, and then substitute the result into the other equation.

Substitution method can be applied in four steps

Solve one of the equations for either x = or y = .

Substitute the solution from step 1 into the other equation.

Solve this new equation.

Solve for the second variable.

Example 1: Solve the following system by substitution

Step 1: Solve one of the equations for either x = or y = . We will solve second equation for y.

Step 2: Substitute the solution from step 1 into the second equation.

Step 3: Solve this new equation.

Step 4: Solve for the second variable

The solution is: (x, y) = (10, -5)

Note: It does not matter which equation we choose first and which second. Just choose the most convenient one first!

Example 2: Solve by substitution

Step 1: Solve one of the equations for either x = or y =. Since the coefficient of y in equation 2 is -1, it is easiest to solve for y in equation 2.

Step 3: Solve this new equation ( for x ).

The solution is: $(x, y) = (1, 2)$

Exercise: Solve the following systems by substitution

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Solving Systems of Equations by Substitution: Explanation, Review, and Examples

  • The Albert Team
  • Last Updated On: February 16, 2023

how to solve problems with substitution

To understand solving systems of equations by substitution, let’s first think about what substitution means.

We use substitution in many situations. First, when the pitcher on a softball team hurts her shoulder, another player can take her place as a substitute. Next, we can substitute vegan alternatives to animal products in a recipe. Finally, when we run into construction on our drive home and take a different road, we’re substituting one route for another. In each of these examples of substitution, we are replacing one entity with another equivalent one to solve a problem or reach a goal.

In the study of Algebra, we learn how to substitute variables for mathematical values in expressions. There is an algebraic property of equality called the Substitution Property , which states:

For example, we can substitute 7 for x in the following equation.

We can do this because 5+2=7 .

The equation above has only one variable. What about situations where we have two or more variables and two or more equations? These  systems of equations  can seem more challenging, but solving systems of linear equations by substitution is often the easiest way to find solutions. 

Solving Systems of Equations Algebraically

There are multiple methods for solving systems of equations, including solving systems algebraically. Solving systems algebraically involves manipulating the equations we are given to uncover the values of each of the variables. 

But when must a system of linear equations be solved algebraically? When solving systems of equations, we should generally choose the method that takes the least effort and leaves the least room for error.

Let’s look at the system of linear equations below:

First, notice we are given the value of one of the variables, x . So, we can easily substitute that value for x into the other equation and solve for y . 

To review how to solve equations, check out our post: Solving One-Step Equations .

When we practice solving systems of equations, students are often told which method to use to find the solution. The directions might say: solve by graphing, solve by elimination, or solve by substitution. We will cover the first two methods in other posts. Here, we will focus on how to solve a system of equations algebraically using substitution.

How to Solve a System of Equations by Substitution

To solve a system of equations by substitution, we can rewrite a two-variable equation as a single variable equation by substituting the value of a variable from one equation into the other.

Let’s start by solving the system of equations that we looked at above:

As we decide how to solve systems of equations with substitution, we almost always have options. We have to decide which variable to substitute and which equation to substitute it into. 

In this example, the choice is clear. Since the first equation says that x=4 , we will substitute x with 4 in the second equation so that the second equation becomes:

Next, we can use the subtraction property of equality to subtract 4 from each side of the equation: 

We just solved this system of linear equations with substitution! The solution to this system is (4,8) .

Knowing that the solution to a system of linear equations is the point of intersection, we can confirm graphically that the coordinate pair (4,8) is the solution to this system of equations.

how to solve problems with substitution

Pro Tip: Online graphing calculators like Desmos can help you check your work quickly and easily.

Solving Systems of Equations by Substitution Steps

So, the steps for using the substitution method to solve a system of linear equations are:

  • Rewrite one of the equations to isolate one variable.
  • In the other equation, substitute the value of your isolated variable in for that variable.
  • Solve this second equation for the other variable. You should have a numerical value.
  • Substitute your numerical value into one of the two original equations and solve for the other variable.
  • Check your work. You can do this by graphing or by substituting the solutions into the original equations.

Substitution Method Examples

It’s helpful to use these steps when we consider how to solve systems of equations by substitution. Now, we can apply these steps to various systems to see if they work.

Solving Systems of Equations by Substitution Examples (One Solution)

Let’s see if these steps work for another system of equations:

1. Rewrite one of the equations to isolate one variable. Let’s solve the first equation for y :

2. In the other equation, substitute the value of your isolated variable in for that variable. So, we will substitute 10-x in for y into the second equation so that it becomes:

3. Solve this second equation for the other variable. In this case, we are solving for x :

4. Substitute your numerical value into one of the two original equations and solve for the other variable. We’ll substitute 6 for x into the first equation and solve for y .

5. Check your work either. This time, we’ll confirm algebraically that the coordinate pair ({\color{red}{6}},{\color{blue}{4}}) works by substituting those values into the other equation, x-y=2 :

Solving Systems of Equations by Substitution Examples (No Solution)

The systems of equations we have solved so far had one solution, but systems of equations may also have zero, multiple, or an infinite number of solutions. Let’s solve a no solution system of equations by substitution:

Notice that y is isolated in the second equation. So, we can substitute (-x+1) for y into the first equation so that it becomes: 

At this point, we have a statement that is not true. A false statement tells us that there is no solution to the system of equations.

If we graph this system, we will see that these are equations of parallel lines, and parallel lines never intersect.

how to solve problems with substitution

Solving Systems of Equations by Substitution Examples (Infinite Solutions)

Let’s solve another system of linear equations by substitution:

In this system, the first equation almost has y isolated, so let’s rewrite that one:

Now we can substitute -x+4 for y in the second equation and solve algebraically:

Our equation is a true statement. However, it doesn’t tell us the values of our variables. Therefore, there are an infinite number of solutions to this system of equations. How can that be? Both equations graph as the same line. We can verify this by rewriting each equation into slope-intercept form.

Earlier, we found that the first equation can be rewritten as:

Next, the second equation becomes:

Since the equations are the same, the lines fully overlap. So, the system will have an infinite number of solutions because there are an infinite number of coordinate pairs that lie on both lines.

The Substitution Method: Keys to Remember

  • Substitution is a helpful strategy in both life and math.
  • Solving systems of equations algebraically involves using the Properties of Algebra.
  • Substitution may be the obvious way to approach a system of equations, or question directions may require using substitution to solve systems of linear equations.
  • Substitution allows us to eliminate one variable in a two-variable equation and solve for the other. 
  • Once the value of one variable is determined, we can use substitution again to solve for the other variable(s).
  • Systems of equations may have zero, one, multiple, or infinite solutions which can be determined and checked algebraically.

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Systems of Linear Equations: Solving by Substitution

Definitions Graphing Special Cases Substitution Elimination/Addition Gaussian Elimination More Examples

The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable.

Here is how it works. (I'll use one of the systems from the " solving by graphing " page.)

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Solving Systems of Equations by Substitution on MathHelp.com

Solving Systems by Substitution

  • Solve the following system by substitution.

2 x − 3 y = −2 4 x + y = 24

The instructions tell me to solve "by substitution". This means that I need to solve one of the equations for one of the variables, and plug the result into the other equation in place of the variable I've solved for. It does not matter which equation or which variable I pick. There is no right or wrong choice of equation or variable; the answer will be the same, regardless. But — some choices may be better than others.

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For instance, in this case, can you see that it would probably be simplest to solve the second equation for " y = ", since there is already a y floating around loose in the middle of that equation? I could solve the first equation for either variable, but I'd get fractions, and solving the second equation for x would also give me fractions. It wouldn't be "wrong" to make a different choice, but it would probably be more difficult.

Being lazy, I'll solve the second equation for y :

4 x + y = 24 y = −4 x + 24

Now I'll plug this in (that is, I'll "substitute " this) for y in the first equation, and solve the resulting one-variable equation for the value of x :

2 x − 3(−4 x + 24) = −2 2 x + 12 x − 72 = −2 14 x = 70 x = 5

Now I can plug this x -value back into either equation, and solve for the corresponding value of y . But since I already have an expression for " y = ", it will be simplest to just plug into this:

y = −4(5) + 24 = −20 + 24 = 4

Then my solution is the following point:

solution: ( x , y ) = (5, 4)

How do you pick the equation to substitute into?

There is no particular rule about which equation to plug into. You picked one of the equations and solved it for one of the variables. You can now pick any of the *other* equations that contains that solved-for variable, and plug in for that variable. The idea is to extract some information from the first equation, and then plug its info into one of the other equations, and see where that takes you.

Both the first equation you solve, and the different equation you plug into, are entirely your choice. There can be many correct ways to solve a system by substitution, but the most important thing to remember is to plug into a different equation rather than the one you'd started with.

For instance, in the above exercise, if I had substituted my " −4 x + 24 " expression into the same equation as I'd used to solve for " y = ", I would have gotten a true, but useless, statement:

4 x + (−4 x + 24) = 24 4 x − 4 x + 24 = 24 24 = 24

Yes, twenty-four does equal twenty-four, but who cares? How does this help? It doesn't, other than to suggest that something might be wrong with your process or assumptions.

So, when using substitution, make sure you substitute from one equation into the other equation, or you'll just be wasting your time.

y = 36 − 9 x 3 x + y /3 = 12

We already know (from the previous page ) that these equations are actually both the same line; that is, that this is a dependent system. We know what this looks like graphically: we get two identical line equations, and we get a graph that displays just the one line. But what does this look like algebraically?

The first equation is already solved for y , so I'll substitute that into the second equation:

3 x + (36 − 9 x )/3 = 12 3 x + 12 − 3 x = 12 12 = 12

Well, um... yes, twelve does equal twelve, but so what?

I did substitute the result from the first equation into the second equation, so this unhelpful result is not because of some screw-up on my part. It's just that this is what a dependent system looks like when you try to find a solution.

Remember that, when you're trying to solve a system, you're trying to use the second equation to narrow down the choices of points on the first equation. You're trying to find the one single point that works in both equations. But in a dependent system, the "second" equation is really just another copy of the first equation, and all the points on the one line will work in the other line.

In other words, I got an unhelpful result because the second line equation didn't tell me anything new. This tells me that the system is actually dependent, and that the solution is the whole line:

solution: y = 36 − 9 x

This is always true, by the way. When you try to solve a system and you get a statement like " 12 = 12 " or " 0 = 0 " — something that's true, but unhelpful (I mean, duh!, of course twelve equals twelve!) — then you have a dependent system. We already knew (from the previous page) that this system was dependent, but now you know what the algebra looks like.

(Keep in mind that your text may format the answer to look something like " ( t , 36 − 9 t ) ", or something similar, using some variable, some "parameter", other than x . But this "parametrized" form of the solution means the exact same thing as "the solution is the line y = 36 − 9 x ".)

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7 x + 2 y = 16 −21 x − 6 y = 24

Neither of these equations is particularly easier than the other for solving. I'll get fractions, no matter which equation and which variable I choose. So, um... I guess I'll take the first equation, and I'll solve it for, um, y , because at least the 2 (from the " 2 y ") will divide evenly into the 16 .

7 x + 2 y = 16 2 y = −7 x + 16 y = −(7/2) x + 8

Now I'll plug this into the other equation:

−21 x − 6(−(7/2) x + 8) = 24 −21 x + 21 x − 48 = 24 −48 = 24

Um... I don't think that's right....

In this case, I got a nonsense result. All my math was right, but I got an obviously wrong answer. So what happened?

Keep in mind that, when solving, you're trying to find where the lines intersect. But what if they don't intersect?

Then you're going to get some kind of wrong answer when you assume that there is a solution (as I did when I tried to find that solution). We knew, from the previous page, that this system represents two parallel lines. But I tried, by substitution, to find the intersection point anyway. And I got a "garbage" result. Since there wasn't any intersection point, my attempt led to utter nonsense.

solution: no solution (inconsistent system)

This is always true, by the way. When you get a nonsense result, this is the algebraic indication that the system of equations is inconsistent.

Note that this is quite different from the previous example:

A true-but-useless result (like " 12 = 12 ") is quite different from a nonsense "garbage" result (like " −48 = 24 "), just as two identical lines are quite different from two distinct parallel lines.

A useless result means a dependent system which has a solution (every point on the whole line, rather than just one point); a nonsense result means an inconsistent system which has no solution of any kind. Don't confuse the two!

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how to solve problems with substitution

Substitution Method

One of the methods to solve a system of linear equations in two variables algebraically is the "substitution method". In this method, we find the value of any one of the variables by isolating it on one side and taking every other term on the other side of the equation. Then we substitute that value in the second equation.

The substitution method is preferable when one of the variables in one of the equations has a coefficient of 1. It involves simple steps to find the values of variables of a system of linear equations by substitution method. Let's learn about it in detail in this article.

What is Substitution Method?

The substitution method is a simple way to solve a system of  linear equations algebraically and find the solutions of the variables. As the name suggests, it involves finding the value of the x-variable in terms of the y-variable from the first equation and then substituting or replacing the value of the x-variable in the second equation. In this way, we can solve and find the value of the y-variable. And at last, we can put the value of y in any of the given equations to find x This process can be interchanged as well where we first solve for x and then solve for y.

In simple words, the substitution method involves substituting the value of any one of the variables from one equation into the other equation. Let us take an example of solving two equations x-2y=8 and x+y=5 using the substitution method.

solving system of linear equations by substitution method gives the result as x equals 6 and y equals minus 1

☛ Note:  The other three algebraic methods of solving linear equations . To learn each of these methods, click on the respective links given below.

  • Elimination method
  • Cross multiplication method
  • Graphical method

Solving Systems of Equations by Substitution Method

The steps to apply or use the substitution method to solve a system of equations are given below:

  • Step 1:  Simplify the given equation by expanding the parenthesis if needed.
  • Step 2: Solve any one of the equations for any one of the variables. You can use any variable based on the ease of calculation.
  • Step 3: Substitute the obtained value of x or y in the other equation.
  • Step 4: Now, simplify the new equation obtained using arithmetic operations  and solve the equation for one variable.
  • Step 5: Now, substitute the value of the variable from  Step 4  in any of the given equations to solve for the other variable.

Here is an example of solving system of equations by using substitution method: 2x+3(y+5)=0 and x+4y+2=0.

Step 1:  Simplify the first equation to get 2x + 3y + 15 = 0. Now we have two equations as,

2x + 3y + 15 = 0 _____ (1)

x + 4y + 2 = 0 ______ (2)

Step 2: We are solving equation (2) for x. So, we get x = -4y - 2.

Step 3: Substitute the obtained value of x in the equation (1). i.e., we are substituting x = -4y-2 in the equation 2x + 3y + 15 = 0, we get, 2(-4y-2) + 3y + 15 = 0.

Step 4: Now, simplify the new equation. We get, -8y-4+3y+15=0

-5y + 11 = 0

Step 5: Now, substitute the value of y in any of the given equations. Let us substitute the value of y in equation (2).

x + 4y + 2 = 0

x + 4 × (11/5) + 2 = 0

x + 44/5 + 2 = 0

x + 54/5 = 0

Therefore, after solving the given system of equations by substitution method, we get x = -54/5 and y= 11/5.

Difference Between Elimination and Substitution Method

Both elimination and substitution methods are ways to solve linear equations algebraically. When the substitution method becomes a little difficult to apply in equations involving large numbers or fractions, we can use the elimination method to ease our calculations. Let us understand the difference between these two methods through the table given below:

Important Notes on Substitution Method:

  • To start with the substitution method, first, select the equation that has coefficient 1 for at least one of the variables and solve for the same variable (with coefficient 1). This makes the process easier.
  • Before starting with the substitution method, combine all like terms (if any).
  • After solving for one variable, we can select any of the given equations or any equation in the whole process to find the other variable.
  • If we get any true statement like 3 = 3, 0 = 0, etc while solving using the substitution method, then it means that the system has infinitely many solutions.
  • If we get any false statement like 3 = 2, 0 = 1, etc then the system has no solution.

☛  Related Topics:

  • Substitution Method Calculator
  • Substitution Method Class 10
  • System of Equations Solver

Substitution Method Examples

Example 1:  Solve the system of linear equations by substitution method: 5m−2n=17 and 3m+n=8.

The given two equations are:

5m−2n=17 ____ (1)

3m+n=8 _____ (2)

The solution of the given two equations can be found by the following steps:

  • From equation 2 we can find the value of n in terms of m, where n = 8 - 3m
  • Substitute the value of n in equation 1. We get, 5m - 2(8-3m)=17

5m - 2(8-3m)=17

5m - 16 + 6m =17

11m = 17 + 16

  • Substitute the value of m in equation 2, we get, 3×3+n=8

Answer:  ∴ The solution is m=3 and n=-1.

Example 2: Jacky has two numbers such that the sum of two numbers is 20 and the difference between them is 10. Find the numbers.

Let the two numbers be x and y such that x>y. It is given that,

x+y=20 ___ (1)

and x−y=10 ___ (2).

We will now solve by substitution.

From equation 1, we get x = 20-y. Substitute this value in equation 2 to find the value of y.

Now, substitute the value of y in equation 1, we get x+5=20, which gives us x=15.

Answer:  Therefore, the two numbers are 15 and 5.

Example 3: Solve the given system of linear equations by substitution method:

- 2x - 5 + 3x + y = 0 ___ (1)

3x + y = 11 ___ (2)

As we can see that the first equation can be further simplified by combining like terms . After simplifying it, we get x+y-5=0. From this equation, let us find the value of x in terms of y, which is x = 5-y. Now substitute this value in equation 2, we get 3(5-y)+y=11.

Now, let us substitute the value of y in equation 1. We get x+2-5=0, which can be simplified to x = 3.

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FAQs on Substitution Method

What is the substitution method in algebra.

In algebra, the substitution method is one of the ways to solve linear equations in two variables . In this method, we substitute the value of a variable found by one equation in the second equation. It is very easy to use when we have smaller numbers, but in the case of large numbers or fractional coefficients, it becomes tedious to apply the substitution method.

What are the Steps for the Substitution Method?

The three simple steps for the substitution method are given below:

  • Find the value of any one variable from any of the equations in terms of the other variable.
  • Substitute it in the other equation and solve.
  • Substitute the value of the second variable again in any of the equations.

When would you Use the Substitution Method?

The substitution method can be applied to any pair of linear equations with two variables . It is advisable to use the substitution method when we have smaller coefficients in terms or when the equations are given in form x = ay+c and/or y=bx+p.

What do we Substitute in the Substitution Method?

In the substitution method, we substitute the value of one variable found by simplifying an equation in the other equation. For example, if there are two variables in the equations, say m and n, then we can first find the value of m in terms of n from any one of the equations, and then we substitute that value in the second equation to get an answer of n. Then, we again substitute the value of n in any of the given equations to find m.

What do the Substitution Method and the Elimination Method have in Common?

Both methods involve the process of substitution. In both methods, we find the value of one variable first and then substitute it in any of the given equations. 

What is the First Step in the Substitution Method?

The first step in the substitution method is to find the value of any one of the variables from one equation in terms of the other variable. For example, if there are two equations x+y=7 and x-y=8, then from the first equation we can find that x=7-y. The further steps involve substituting this in the other equation and then solving.

What is the Process of Solving Systems by Substitution?

With a system of equations with variables x and y, we first find the value of x in terms of y from any one of the equations given. Then, we substitute that value in the other equation to find the value of y. At last, we again substitute the value of y in any given equation to find x.

Is the Substitution Method Only for Linear Equations?

No, substitution method can be applied for any type of equations . For example, the equations y = x 2  and y = 3x + 4 can be solved by using the substitution method.

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  • 8.7 Use Radicals in Functions
  • 8.8 Use the Complex Number System
  • 9.1 Solve Quadratic Equations Using the Square Root Property
  • 9.2 Solve Quadratic Equations by Completing the Square
  • 9.3 Solve Quadratic Equations Using the Quadratic Formula
  • 9.4 Solve Equations in Quadratic Form
  • 9.5 Solve Applications of Quadratic Equations
  • 9.6 Graph Quadratic Functions Using Properties
  • 9.7 Graph Quadratic Functions Using Transformations
  • 9.8 Solve Quadratic Inequalities
  • 10.1 Finding Composite and Inverse Functions
  • 10.2 Evaluate and Graph Exponential Functions
  • 10.3 Evaluate and Graph Logarithmic Functions
  • 10.4 Use the Properties of Logarithms
  • 10.5 Solve Exponential and Logarithmic Equations
  • 11.1 Distance and Midpoint Formulas; Circles
  • 11.2 Parabolas
  • 11.3 Ellipses
  • 11.4 Hyperbolas
  • 12.1 Sequences
  • 12.2 Arithmetic Sequences
  • 12.3 Geometric Sequences and Series
  • 12.4 Binomial Theorem

Learning Objectives

By the end of this section, you will be able to:

  • Solve a system of nonlinear equations using graphing
  • Solve a system of nonlinear equations using substitution
  • Solve a system of nonlinear equations using elimination
  • Use a system of nonlinear equations to solve applications

Be Prepared 11.13

Solve the system by graphing: { x − 3 y = −3 x + y = 5 . { x − 3 y = −3 x + y = 5 . If you missed this problem, review Example 4.2 .

Be Prepared 11.14

Solve the system by substitution: { x − 4 y = −4 − 3 x + 4 y = 0 . { x − 4 y = −4 − 3 x + 4 y = 0 . If you missed this problem, review Example 4.7 .

Be Prepared 11.15

Solve the system by elimination: { 3 x − 4 y = −9 5 x + 3 y = 14 . { 3 x − 4 y = −9 5 x + 3 y = 14 . If you missed this problem, review Example 4.9 .

Solve a System of Nonlinear Equations Using Graphing

We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A system of nonlinear equations is a system where at least one of the equations is not linear.

For example each of the following systems is a system of nonlinear equations .

System of Nonlinear Equations

A system of nonlinear equations is a system where at least one of the equations is not linear.

Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.

When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.

To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.

Solve a system of nonlinear equations by graphing.

  • Step 1. Identify the graph of each equation. Sketch the possible options for intersection.
  • Step 2. Graph the first equation.
  • Step 3. Graph the second equation on the same rectangular coordinate system.
  • Step 4. Determine whether the graphs intersect.
  • Step 5. Identify the points of intersection.
  • Step 6. Check that each ordered pair is a solution to both original equations.

Example 11.33

Solve the system by graphing: { x − y = −2 y = x 2 . { x − y = −2 y = x 2 .

Try It 11.65

Solve the system by graphing: { x + y = 4 y = x 2 + 2 . { x + y = 4 y = x 2 + 2 .

Try It 11.66

Solve the system by graphing: { x − y = −1 y = − x 2 + 3 . { x − y = −1 y = − x 2 + 3 .

To identify the graph of each equation, keep in mind the characteristics of the x 2 x 2 and y 2 y 2 terms of each conic.

Example 11.34

Solve the system by graphing: { y = −1 ( x − 2 ) 2 + ( y + 3 ) 2 = 4 . { y = −1 ( x − 2 ) 2 + ( y + 3 ) 2 = 4 .

Try It 11.67

Solve the system by graphing: { x = −6 ( x + 3 ) 2 + ( y − 1 ) 2 = 9 . { x = −6 ( x + 3 ) 2 + ( y − 1 ) 2 = 9 .

Try It 11.68

Solve the system by graphing: { y = 4 ( x − 2 ) 2 + ( y + 3 ) 2 = 4 . { y = 4 ( x − 2 ) 2 + ( y + 3 ) 2 = 4 .

Solve a System of Nonlinear Equations Using Substitution

The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.

The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.

Solve a system of nonlinear equations by substitution.

  • Step 2. Solve one of the equations for either variable.
  • Step 3. Substitute the expression from Step 2 into the other equation.
  • Step 4. Solve the resulting equation.
  • Step 5. Substitute each solution in Step 4 into one of the original equations to find the other variable.
  • Step 6. Write each solution as an ordered pair.
  • Step 7. Check that each ordered pair is a solution to both original equations.

Example 11.35

Solve the system by using substitution: { 9 x 2 + y 2 = 9 y = 3 x − 3 . { 9 x 2 + y 2 = 9 y = 3 x − 3 .

Try It 11.69

Solve the system by using substitution: { x 2 + 9 y 2 = 9 y = 1 3 x − 3 . { x 2 + 9 y 2 = 9 y = 1 3 x − 3 .

Try It 11.70

Solve the system by using substitution: { 4 x 2 + y 2 = 4 y = x + 2 . { 4 x 2 + y 2 = 4 y = x + 2 .

So far, each system of nonlinear equations has had at least one solution. The next example will show another option.

Example 11.36

Solve the system by using substitution: { x 2 − y = 0 y = x − 2 . { x 2 − y = 0 y = x − 2 .

Try It 11.71

Solve the system by using substitution: { x 2 − y = 0 y = 2 x − 3 . { x 2 − y = 0 y = 2 x − 3 .

Try It 11.72

Solve the system by using substitution: { y 2 − x = 0 y = 3 x − 2 . { y 2 − x = 0 y = 3 x − 2 .

Solve a System of Nonlinear Equations Using Elimination

When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.

The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.

Solve a system of equations by elimination.

  • Step 2. Write both equations in standard form.
  • Step 3. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites.
  • Step 4. Add the equations resulting from Step 3 to eliminate one variable.
  • Step 5. Solve for the remaining variable.
  • Step 6. Substitute each solution from Step 5 into one of the original equations. Then solve for the other variable.
  • Step 7. Write each solution as an ordered pair.
  • Step 8. Check that each ordered pair is a solution to both original equations.

Example 11.37

Solve the system by elimination: { x 2 + y 2 = 4 x 2 − y = 4 . { x 2 + y 2 = 4 x 2 − y = 4 .

Try It 11.73

Solve the system by elimination: { x 2 + y 2 = 9 x 2 − y = 9 . { x 2 + y 2 = 9 x 2 − y = 9 .

Try It 11.74

Solve the system by elimination: { x 2 + y 2 = 1 − x + y 2 = 1 . { x 2 + y 2 = 1 − x + y 2 = 1 .

There are also four options when we consider a circle and a hyperbola.

Example 11.38

Solve the system by elimination: { x 2 + y 2 = 7 x 2 − y 2 = 1 . { x 2 + y 2 = 7 x 2 − y 2 = 1 .

Try It 11.75

Solve the system by elimination: { x 2 + y 2 = 25 y 2 − x 2 = 7 . { x 2 + y 2 = 25 y 2 − x 2 = 7 .

Try It 11.76

Solve the system by elimination: { x 2 + y 2 = 4 x 2 − y 2 = 4 . { x 2 + y 2 = 4 x 2 − y 2 = 4 .

Use a System of Nonlinear Equations to Solve Applications

Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.

Example 11.39

The difference of the squares of two numbers is 15. The sum of the numbers is 5. Find the numbers.

Try It 11.77

The difference of the squares of two numbers is −20 . −20 . The sum of the numbers is 10. Find the numbers.

Try It 11.78

The difference of the squares of two numbers is 35. The sum of the numbers is −1 . −1 . Find the numbers.

Example 11.40

Myra purchased a small 25” TV for her kitchen. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 300 square inches. What are the length and width of the TV screen?

Try It 11.79

Edgar purchased a small 20” TV for his garage. The size of a TV is measured on the diagonal of the screen. The screen also has an area of 192 square inches. What are the length and width of the TV screen?

Try It 11.80

The Harper family purchased a small microwave for their family room. The diagonal of the door measures 15 inches. The door also has an area of 108 square inches. What are the length and width of the microwave door?

Access these online resources for additional instructions and practice with solving nonlinear equations.

  • Nonlinear Systems of Equations
  • Solve a System of Nonlinear Equations
  • Solve a System of Nonlinear Equations by Elimination
  • System of Nonlinear Equations – Area and Perimeter Application

Section 11.5 Exercises

Practice makes perfect.

In the following exercises, solve the system of equations by using graphing.

{ y = 2 x + 2 y = − x 2 + 2 { y = 2 x + 2 y = − x 2 + 2

{ y = 6 x − 4 y = 2 x 2 { y = 6 x − 4 y = 2 x 2

{ x + y = 2 x = y 2 { x + y = 2 x = y 2

{ x − y = −2 x = y 2 { x − y = −2 x = y 2

{ y = 3 2 x + 3 y = − x 2 + 2 { y = 3 2 x + 3 y = − x 2 + 2

{ y = x − 1 y = x 2 + 1 { y = x − 1 y = x 2 + 1

{ x = −2 x 2 + y 2 = 4 { x = −2 x 2 + y 2 = 4

{ y = −4 x 2 + y 2 = 16 { y = −4 x 2 + y 2 = 16

{ x = 2 ( x + 2 ) 2 + ( y + 3 ) 2 = 16 { x = 2 ( x + 2 ) 2 + ( y + 3 ) 2 = 16

{ y = −1 ( x − 2 ) 2 + ( y − 4 ) 2 = 25 { y = −1 ( x − 2 ) 2 + ( y − 4 ) 2 = 25

{ y = −2 x + 4 y = x + 1 { y = −2 x + 4 y = x + 1

{ y = − 1 2 x + 2 y = x − 2 { y = − 1 2 x + 2 y = x − 2

In the following exercises, solve the system of equations by using substitution.

{ x 2 + 4 y 2 = 4 y = 1 2 x − 1 { x 2 + 4 y 2 = 4 y = 1 2 x − 1

{ 9 x 2 + y 2 = 9 y = 3 x + 3 { 9 x 2 + y 2 = 9 y = 3 x + 3

{ 9 x 2 + y 2 = 9 y = x + 3 { 9 x 2 + y 2 = 9 y = x + 3

{ 9 x 2 + 4 y 2 = 36 x = 2 { 9 x 2 + 4 y 2 = 36 x = 2

{ 4 x 2 + y 2 = 4 y = 4 { 4 x 2 + y 2 = 4 y = 4

{ x 2 + y 2 = 169 x = 12 { x 2 + y 2 = 169 x = 12

{ 3 x 2 − y = 0 y = 2 x − 1 { 3 x 2 − y = 0 y = 2 x − 1

{ 2 y 2 − x = 0 y = x + 1 { 2 y 2 − x = 0 y = x + 1

{ y = x 2 + 3 y = x + 3 { y = x 2 + 3 y = x + 3

{ y = x 2 − 4 y = x − 4 { y = x 2 − 4 y = x − 4

{ x 2 + y 2 = 25 x − y = 1 { x 2 + y 2 = 25 x − y = 1

{ x 2 + y 2 = 25 2 x + y = 10 { x 2 + y 2 = 25 2 x + y = 10

In the following exercises, solve the system of equations by using elimination.

{ x 2 + y 2 = 16 x 2 − 2 y = 8 { x 2 + y 2 = 16 x 2 − 2 y = 8

{ x 2 + y 2 = 16 x 2 − y = 4 { x 2 + y 2 = 16 x 2 − y = 4

{ x 2 + y 2 = 4 x 2 + 2 y = 1 { x 2 + y 2 = 4 x 2 + 2 y = 1

{ x 2 + y 2 = 4 x 2 − y = 2 { x 2 + y 2 = 4 x 2 − y = 2

{ x 2 + y 2 = 9 x 2 − y = 3 { x 2 + y 2 = 9 x 2 − y = 3

{ x 2 + y 2 = 4 y 2 − x = 2 { x 2 + y 2 = 4 y 2 − x = 2

{ x 2 + y 2 = 25 2 x 2 − 3 y 2 = 5 { x 2 + y 2 = 25 2 x 2 − 3 y 2 = 5

{ x 2 + y 2 = 20 x 2 − y 2 = −12 { x 2 + y 2 = 20 x 2 − y 2 = −12

{ x 2 + y 2 = 13 x 2 − y 2 = 5 { x 2 + y 2 = 13 x 2 − y 2 = 5

{ x 2 + y 2 = 16 x 2 − y 2 = 16 { x 2 + y 2 = 16 x 2 − y 2 = 16

{ 4 x 2 + 9 y 2 = 36 2 x 2 − 9 y 2 = 18 { 4 x 2 + 9 y 2 = 36 2 x 2 − 9 y 2 = 18

{ x 2 − y 2 = 3 2 x 2 + y 2 = 6 { x 2 − y 2 = 3 2 x 2 + y 2 = 6

{ 4 x 2 − y 2 = 4 4 x 2 + y 2 = 4 { 4 x 2 − y 2 = 4 4 x 2 + y 2 = 4

{ x 2 − y 2 = −5 3 x 2 + 2 y 2 = 30 { x 2 − y 2 = −5 3 x 2 + 2 y 2 = 30

{ x 2 − y 2 = 1 x 2 − 2 y = 4 { x 2 − y 2 = 1 x 2 − 2 y = 4

{ 2 x 2 + y 2 = 11 x 2 + 3 y 2 = 28 { 2 x 2 + y 2 = 11 x 2 + 3 y 2 = 28

In the following exercises, solve the problem using a system of equations.

The sum of two numbers is −6 −6 and the product is 8. Find the numbers.

The sum of two numbers is 11 and the product is −42 . −42 . Find the numbers.

The sum of the squares of two numbers is 65. The difference of the numbers is 3. Find the numbers.

The sum of the squares of two numbers is 113. The difference of the numbers is 1. Find the numbers.

The difference of the squares of two numbers is 15. The difference of twice the square of the first number and the square of the second number is 30. Find the numbers.

The difference of the squares of two numbers is 20. The difference of the square of the first number and twice the square of the second number is 4. Find the numbers.

The perimeter of a rectangle is 32 inches and its area is 63 square inches. Find the length and width of the rectangle.

The perimeter of a rectangle is 52 cm and its area is 165 cm 2 . cm 2 . Find the length and width of the rectangle.

Dion purchased a new microwave. The diagonal of the door measures 17 inches. The door also has an area of 120 square inches. What are the length and width of the microwave door?

Jules purchased a microwave for his kitchen. The diagonal of the front of the microwave measures 26 inches. The front also has an area of 240 square inches. What are the length and width of the microwave?

Roman found a widescreen TV on sale, but isn’t sure if it will fit his entertainment center. The TV is 60”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1728 square inches. His entertainment center has an insert for the TV with a length of 50 inches and width of 40 inches. What are the length and width of the TV screen and will it fit Roman’s entertainment center?

Donnette found a widescreen TV at a garage sale, but isn’t sure if it will fit her entertainment center. The TV is 50”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of 1200 square inches. Her entertainment center has an insert for the TV with a length of 38 inches and width of 27 inches. What are the length and width of the TV screen and will it fit Donnette’s entertainment center?

Writing Exercises

In your own words, explain the advantages and disadvantages of solving a system of equations by graphing.

Explain in your own words how to solve a system of equations using substitution.

Explain in your own words how to solve a system of equations using elimination.

A circle and a parabola can intersect in ways that would result in 0, 1, 2, 3, or 4 solutions. Draw a sketch of each of the possibilities.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • Authors: Lynn Marecek, Andrea Honeycutt Mathis
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  • Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
  • Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/11-5-solve-systems-of-nonlinear-equations

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The Substitution Method

A way to solve systems of linear equations in 2 variables

Video on Solving by Substitution

First, let's review how the substitution property works in general.

Review Example 1

substitution property example 1

Review Example 2

substitution property example 2

Substitution Example 1

picture of algebraic method solution

Let's re-examine system pictured up above.

$ \red{y} = 2x + 1 \text{ and } \red{y} = 4x -1 $

We are going to use substitution like we did in review example 2 above.

example 3

Now we have 1 equation and 1 unknown, we can solve this problem as the work below shows.

The last step is to again use substitution, in this case we know that x = 1, but in order to find the y value of the solution, we just substitute x = 1 into either equation.

$$ y = 2x + 1 \\ y = 2\cdot \red{1} + 1 = 2 + 1 =3 \\ \\ \boxed{ \text{ or you use the other equation}} \\ y = 4x -1 \\ y = 4\cdot \red{1}- 1 \\ y = 4 - 1 = 3 \\ \boxed { ( 1,3) } $$

Substitution Example 2

What is the solution of the system of equations below:

$ y = 2x + 1 \\ 2y = 3x - 2 $

Identify the best equation for substitution and then substitute into other equation.

example 3

Solve for x

Substitute the value of x (-4 in this case) into either equation.

$$ y = 2x + 1 \\ y = 2\cdot \red{-4} + 1 = -8 + 1 = -7 \\ 2y = 3x - 2\\ 2y = 3\cdot-4 -2 \\ \boxed{ \text{ or you use the other equation}} \\ 2y = 3x -2 \\ 2y = 3 ( \red{-4}) -2 \\ 2y = -12 -2 \\ 2y = -14 \frac{1}{2}\cdot2y =\frac{1}{2}\cdot-14 \\ y = -7 $$

$$ \boxed { ( -4, -7 ) } $$

You can also solve the system by graphing and see a picture of the solution below:

Double Check Substitution Method

Substitution Practice Problems

Solve the system below using substitution

$$ y = x+1 \\ y = 2x +2 $$

The solution of this system is the point of intersection: (-1, 0).

$$ y = x + 1 \quad y = 2x + 2 \\ \hspace{1.2cm} \downarrow \hspace{1.4cm} \downarrow \\ \hspace{6mm} x + 1 = 2x + 2 \\ \hspace{7mm} \text{-}x \hspace{1.4cm} \text{-}x \\ \hspace{7mm} \rule{3.2cm}{0.25mm} \\ \hspace{1.7cm} 1 = x + 2 \\ \hspace{1.6cm} \text{-}2 \hspace{1.4cm} \text{-}2 \\ \hspace{7mm} \rule{3.2cm}{0.25mm} \\ \hspace{1.2cm} -1 = x \\ \hspace{1.6cm} \downarrow \\ \hspace{5mm} y = 2x + 2 \\ \hspace{7mm} y = 2 * (-1) + 2 = 0 \\[5mm] \text{Solution:} \hspace{3mm} (-1, 0) $$

Use substitution to solve the following system of linear equations:

  • Line 1: y = 3x – 1
  • Line 2: y = x – 5

Set the Two Equations equal to each other then solve for x

Substitute the x value, -2, into the value for 'x' for either equation to determine y coordinate of solution

$$ y = \red{x} -5 \\ y = \red{-2} -5 = -7 $$

The solution is the point (-2, -7)

Use the substitution method to solve the system:

  • Line 1: y = 5x – 1
  • Line 2: 2y= 3x + 12

Solution of system of equations by substitution method

This system of lines has a solution at the point (2, 9).

Use substitution to solve the system:

  • Line 1: y = 3x + 1
  • Line 2: 4y = 12x + 4

This system has an infinite number of solutions. Because 12x + 4 = 12x is always true for all values of x.

Solve the system of linear equations by substitution

  • Line 1: y= x + 2
  • Line 2: y= x + 8

This system of linear equation has no solution .

These lines have the same slope (slope = 1) so they never intersect.

  • Line 1: y = x + 1
  • Line 2: 2y = 3x

Solution answer

The solution of this system is (1, 3).

  • Line 2: 4y = 12x + 3

No Solutions of system of equations by substitution method

Whenever you arrive at a contradiction such as 3 = 4, your system of linear equations has no solutions. When you use these methods (substitution, graphing , or elimination ) to find the solution what you're really asking is at what

Solve the system using substitution.

  • Line 1: y = x + 5
  • Line 2: y = 2x + 2

Practice Problem seven  solution of system of equations

The solution of this system is the point of intersection: (3, 8).

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

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Enter the system of equations you want to solve for by substitution.

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How to Use Substitution to Solve a System of Equations: Word Problems

Solving systems of equations is a fundamental concept in algebra, and it plays a crucial role in various fields such as engineering, physics, and economics. One popular method to solve these systems is substitution, which involves substituting one variable with an expression from another equation. This article aims to provide a comprehensive guide on using substitution to solve a system of equations, particularly focusing on word problems.

How to Use Substitution to Solve a System of Equations: Word Problems

A Step-by-step Guide to Using Substitution to Solve a System of Equations: Word Problems

Let’s go through a step-by-step guide on how to approach these problems.

Step 1: Understand the Problem

Start by carefully reading the word problem. Identify the unknowns and what you need to solve for.

Example Problem: A movie theater sells adult tickets for \($10\) and children tickets for \($6\). If the theater sold \(100\) tickets total and made \($800\), how many adult tickets and children tickets were sold?

The unknowns here are the number of adult and children tickets sold, and we need to find these values.

Step 2: Define the Variables

Next, define variables to represent the unknowns.

For our example:

  • Let’s let ‘\(A\)’ represent the number of adult tickets sold.
  • Let’s let ‘\(C\)’ represent the number of children’s tickets sold.

Step 3: Formulate the Equations

Translate the information from the word problem into mathematical equations.

In our example, we can formulate two equations from the problem:

  • \(A + C = 100\) (The total number of tickets sold is \(100\))
  • \($10A + $6C = $800\) (The total revenue from the tickets is \($800\))

Step 4: Solve One Equation for One Variable

The next step is to solve one of the equations for one variable. It’s usually easier to choose the equation and variable that will be the simplest to solve. In this case, we can solve the first equation for ‘\(A\)’:

\(A = 100 – C\)

Step 5: Substitute the Expression into the Other Equation

Now that we have ‘\(A\)’ in terms of ‘\(C\)’, substitute this expression into the other equation (in place of ‘\(A\)’):

\($10(100 – C) + $6C = $800\)

\($1000 – $10C + $6C = $800\)

\($4C = $100\)

Step 6: Solve for the Other Variable

Substitute \(C= 50\) into the first equation to solve for ‘\(A\)’:

\(A + 50 = 100\)

Step 7: Check Your Answer

Always check your answer to ensure it makes sense in the context of the problem. Substitute \(A= 50\) and \(C= 50\) into both original equations to verify that they are correct:

  • \(50 + 50 = 100\) (This is correct)
  • \($10(50) + $6(50) = $800\) (This is also correct)

Therefore, the theater sold \(50\) adult tickets and \(50\) children’s tickets.

This is a basic walkthrough of using substitution to solve a system of equations. Keep in mind that some problems may be more complex and require additional steps, but the fundamental process is the same.

by: Effortless Math Team about 8 months ago (category: Articles )

Effortless Math Team

Related to this article, more math articles.

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Solving by Substitution

Let's start with a problem that's half done already...  We already know what y is:

We just need to figure out what the x is.  Substitution!  Take the y guy and stick it into the first equation:

Let's double-check that: 

Remember that this is a point where two lines intersect.

Solve by substitution:

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Mathematics LibreTexts

2.4: Solving Differential Equations by Substitutions

  • Last updated
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  • Page ID 103474

  • William F. Trench
  • Trinity University

Bernoulli Equations

Definition 2.4.1.

A Bernoulli equation is an equation of the form

\[\label{eq:2.4.2} {dy\over dx}+p(x)y=f(x)y^r,\]

where \(r\) can be any real number other than \(0\) or \(1\). (Note that Equation \ref{eq:2.4.2} is linear if and only if \(r=0\) or \(r=1\).)

Theorem \(\PageIndex{1}\)

The substitution \(u=y^{1-r},\) will turn the Bernoulli Equation \ref{eq:2.4.2} into a linear equation.

Dividing \ref{eq:2.4.2} by \(y^r\) yields \[y^{-r}{dy\over dx}+p(x)y^{1-r}=f(x).\nonumber\]

If we make the substitution \[u=y^{1-r}\nonumber\] and differentiate with respect to x we get \[{du\over dx}=(1-r)y^{-r}{dy\over dx.}\nonumber\]

From this we can see that \ref{eq:2.4.2} becomes \[{1\over 1-r}{du\over dx}+p(x)u=f(x)\nonumber\] which is linear and can be solved by the methods of section 2.3.

Example 2.4.1

Solve the Bernoulli equation

\[\label{eq:2.4.3} y'-y=xy^2.\]

Dividing by \(y^2\) we get

\[y^{-2}y'-y^{-1}=x.\nonumber\]

If we now let \(u=y^{-1},\) we get \({du\over dx}=-y^{-2}{dy\over dx}\) and substituting into \ref{eq:2.4.3} we get

\[{-du\over dx}-u=x\nonumber\]

which is a linear equation. Putting it in standard linear form we get

\[{du\over dx}+u=-x\nonumber\]

and using the method of section 2.3 we get the solution

\[u=-{(x-1)+ce^{-x}} \nonumber\]

and substituting back in for u we get

\[y^{-1}=-{(x-1)+ce^{-x}} \nonumber\]

and solving for y we get our final solution

\[y=-{1\over x-1+ce^{-x}}. \nonumber\]

Figure 2.4.1 shows the direction field and some integral curves of Equation \ref{eq:2.4.3}.

imageedit_43_4180397678.png

Homogeneous Equations

Definition 2.4.2.

We say that a function \(f(x,y)\) is homogeneous of degree n if

\[f(tx,ty)=t^nf(x,y)\nonumber\]

Example \(\PageIndex{2}\)

Determine if \[f(x,y)=x^5+x^2y^3\nonumber\] is a homogeneous function and, if so, of what degree.

Substituting \(tx\) for \(x\) and \(ty\) for \(y\) we get \[f(tx,ty)=(tx)^5+(tx)^2(ty)^3=t^5(x^5+x^2y^3)=t^5f(x,y).\nonumber\] Therefore, \(f(x,y)\) is a homogeneous function of degree 5.

Example \(\PageIndex{3}\)

Determine if \[f(x,y)=x^2+xy^2\nonumber\] is a homogeneous function and, if so, of what degree.

Substituting \(tx\) for \(x\) and \(ty\) for \(y\) we get \[f(tx,ty)=(tx)^2+(tx)(ty)^2=t^2(x^2+txy^2)\ne t^nf(x,y).\nonumber\] for any number \(n\). Therefore, \(f(x,y)\) is not a homogeneous function.

Note: For this type of differential equation, as in section 2.2, it is convenient to write first order differential equations in the form

\[\label{eq:2.5.1} M(x,y)\,dx+N(x,y)\,dy=0.\]

Definition 2.4.3

We say that a differential equation of the form \(M(x,y)\,dx+N(x,y)\,dy=0\) is homogeneous if \(M(x,y)\) and \(N(x,y)\) are homogeneous functions of the same degree.

Example \(\PageIndex{4}\)

Determine if \[(x^2+xy)dx+y^2dy=0\nonumber\] is a homogeneous differential equation.

Substituting \(tx\) for \(x\) and \(ty\) for \(y\) in \(M(x,y)\) we get \[M(tx,ty)=(tx)^2+(tx)(ty)=t^2(x^2+xy)= t^2M(x,y)\nonumber\] and \(M(x,y)\) is a homogeneous function of degree 2.

Substituting \(tx\) for \(x\) and \(ty\) for \(y\) in \(N(x,y)\) we get \[N(tx,ty)=(ty)^2=t^2y^2= t^2N(x,y)\nonumber\] and \(N(x,y)\) is a homogeneous function of degree 2.

Since both \(M(x,y)\) and \(N(x,y)\) are homogeneous functions of degree 2, the differential equation is homogeneous.

Theorem \(\PageIndex{2}\)

The substitution \(y=ux\), where \(u\) is a function of \(x\), or \(x=vy\), where \(v\) is a function of \(y\) will make a homogeneous equation separable.

Note: we will only prove this for the substitution \(y=ux\) and leave the substitution \(x=vy\) to the reader.

Letting \[y=ux\nonumber\] and differentiating with respect to \(x\) we get

\[{dy\over dx}=u+x{du\over dx}\nonumber\] which can be rewritten as \[dy=udx+xdu.\nonumber\]

Substituting \(y\) and \(dy\) into \ref{eq:2.5.1} we get \[M(x,ux)dx+N(x,ux)[udx+xdu]=0.\nonumber\]

Using the fact that both \(M(x,y)\) and \(N(x,y)\) are homogeneous of the same degree we get

\[x^nM(1,u)dx+x^nN(1,u)[udx+xdu]=0.\nonumber\]

Expanding and rewriting this we eventually come to

\[x^n[M(1,u)+uN(1,u)]dx=-x^{n+1}N(1,u)du\nonumber\] which is clearly a separable differential equation:

\[{1\over x}dx={-N(1,u)\over M(1,u)+uN(1,u)}du.\nonumber\]

Example 2.4.5

\[\label{eq:2.4.8} xdy=({y+xe^{-y/x}})dx.\]

First verify that \(M(x,y)\) and \(N(x,y)\) are homogeneous functions of the same degree.

Substituting \(y=ux\) into Equation \ref{eq:2.4.8} yields

\[x(udx+xdu)= ({ux+xe^{-ux/x}})dx.\nonumber\]

Simplifying and separating variables yields

\[e^udu={1\over x}dx. \nonumber\]

Integrating yields \(e^u=\ln |x|+c\). Therefore \(u=\ln(\ln|x|+c)\) and \(y=ux=x \ln (\ln |x|+c)\).

Figure 2.4.2 shows a direction field and integral curves for Equation \ref{eq:2.4.8}.

imageedit_46_4767318021.png

Equations of the form \({dy\over dx}=f(Ax+By+C)\)

Theorem \(\pageindex{3}\).

The substitution \[u=Ax+By+C\nonumber\] will make equations of the form \[\label{eq:2.4.9} {dy\over dx}=f(Ax+By+C)\] separable.

Consider a differential equation of the form \ref{eq:2.4.9}.

Let \[u=Ax+By+C\nonumber\]

Taking the derivative with respect to x we get \[{du\over dx}=A+B{dy\over dx}.\nonumber\]

Substituting into \ref{eq:2.4.9} we get \[{1\over B}({du\over dx}-A)=f(u)\nonumber\]

which is clearly a separable differential equation: \[{du\over Bf(u)+A}=dx\nonumber\]

Example 2.4.6

\[\label{eq:2.4.10} y'=sec^2(y-x-3).\]

Letting \[u=y-x-3\nonumber\] and differentiating we get \[{du\over dx}={dy\over dx}-1\nonumber\]

Substituting into Equation \ref{eq:2.4.10} we get

\[{du\over dx}+1=sec^2u\nonumber\]

This is now a separable equation which can be solved using the method of section 2.1, resulting in a solution of

\[-\cot u-u=x+c\nonumber\]

and substituting in for u we get

\[-\cot (y-x-3)-(y-x-3)=x+c\nonumber\] and simplifying we get

\[\cot (y-x-3)=-y+c\nonumber\]

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1 Trending: Our Most Serious National Security Threat Isn’t Russian Nukes In Space, It’s Intelligence Agencies In Washington

2 trending: elizabeth warren wants to file your taxes, 3 trending: did our intelligence agencies suggest the russia hoax to hillary clinton’s campaign, 4 trending: president joe biden’s border invasion victimizes young and old americans everywhere, with biden in mental decline, how do you solve a problem like kamala harris.

how to solve problems with substitution

With Biden’s mental acuity becoming an undeniable problem, Democrats will have to figure out what to do with a vice president who’s even more unpopular than him.

Author Mark Hemingway profile

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With yesterday’s special counsel report providing independent confirmation that Joe Biden is probably senile , we need to do something very painful that the Biden campaign and our compliant political press don’t want to do: Have a forthright conversation about Kamala Harris.

Last weekend, NBC News put out a poll that set off klaxons among Democrats. Trump is leading Biden by five points in NBC’s poll, and, more generally, the 2024 general election polls thus far show Trump consistently winning . This is a marked reversal from 2020, where the polls showed Biden with a 7-point lead heading into an election he barely edged out an electoral college victory by 40,000 or so votes.

Now, there are a number of obvious reasons why Biden is flailing this time around. The NBC poll notes Trump has a 20-point advantage on the economy, a 30-point advantage on immigration and border security, and a 16-point advantage on “being competent and effective.” (Democrats are probably pretty despondent to consider he’s being walloped on that last metric by President Covfefe.)

But when you look at Biden’s approval rating in the poll, something else jumps out. Biden’s approval rating – 36 percent approve, 54 percent disapprove of the job he’s doing as president – is absolutely terrible for an incumbent president during an election year. But the NBC poll offers up the approval ratings of all the major candidates, and it turns out Biden doesn’t have the lowest approval rating in the White House. That belongs to Kamala Harris, who is at 28 percent approval, 53 percent disapproval. WOOF.

Now, obviously, given the major issues dragging Biden down, it would hardly seem like Kamala Harris would be the singular thing that sinks Biden, even if her approval rating is comparable to that of venereal disease. However, Biden needs all the help he can get, and given his unique problems, an energetic and engaged vice president would go a long way toward counteracting the negative perception of him.

Obviously, there’s the issue of Biden’s age and cognitive decline. Especially after yesterday’s news, Democrats are not going to get through another election cycle berating reporters that Biden’s confusion is a result of a heroic lifelong struggle to overcome a childhood stutter. Biden is speaking in the present tense about world leaders who died when grunge was still popular , and that’s when he’s capable of speaking at all . Voters can clearly see that mentally, the wheel may be turning, but the hamster is deceased.

As a result, the Biden campaign’s goal is to minimize his public presence during the election, which can only hurt him. They did this to a large extent in 2020, but Covid was a convenient excuse. Now, what’s the reason why Biden is turning down an audience of 20 million people to conduct an interview on Super Bowl Sunday? CNN assures us that turning down such a huge audience is part of a “larger media strategy” where Biden’s advisers “give the already fatigued public a break from politics during the big game.” The reality as we know it is that they simply don’t want to have another viral clip where he’s asked a tough question about Gaza and he responds by nodding off on camera and dribbling cerebral spinal fluid out of the corner of his mouth.

In fact, an AP story from earlier this week, “ Biden is going small to try to win big in November. That means stops for boba tea, burgers and beer ,” does indeed confirm that the Biden strategy for the coming election is, to the extent he’s going to campaign at all, he’s going to do it in intimate, easily controlled settings. They’re scared to let voters interact with him and see him up close.

Now, given that’s the case, just imagine Biden had an energetic, well-liked vice president out on the trail. Voters could at least tell themselves that if they vote for Biden and we have to confront the all-too-likely possibility Biden goes face down in a bowl of Chunky Monkey in the Oval and never wakes up, at least there’s a commanding presence in the wings waiting to step-up.

Instead, we have Kamala Harris, who seems to defy the laws of political physics by existing in two categories at the same time — she’s both actively disliked by voters and an almost complete nonentity when it comes to exerting any influence on policy or politics. I mean, what has she done of note as vice president? Anything at all? During his first year in office, Biden tasked her with overseeing diplomatic efforts allegedly aimed at stopping the mass influx of illegal immigrants across the southern border. How’s that going? Let’s check the spin at CNN :

Since being tasked with tackling root causes, Harris has only occasionally talked about the effort as the situation along the US-Mexico border became a political vulnerability for Biden. … A senior administration official recognized the attention on the US-Mexico border but maintained that Harris’ work is not intended to solve the immediate issues on the ground there.

In other words, Kamala is trying to distance herself from her own responsibilities on an issue that is now voters’ number one concern. “Not intended to solve the immediate issues on the ground there”? What an awe-inspiring display of leadership.

Then there’s the issue of Kamala herself; Biden’s age-related senility means that his verbal stumbles at least induce some measure of pity along with the embarrassment. Kamala’s furor loquendi, on the other hand, well, what the hell are we supposed to make of the fact that even The New York Times concedes , “the vice president’s critics have not exactly fabricated, ex nihilo, the notion that she chops language into what they call ‘word salads.’” Frankly, that’s a polite read on authentic California self-actualized airhead gibberish such as, “I think it’s very important, as you have heard from so many incredible leaders, for us, at every moment in time, and certainly this one, to see the moment in time in which we exist and are present, and to be able to contextualize it, to understand where we exist in the history and in the moment, as it relates not only to the past but the future.”

Then, something must be said about Kamala Harris’ truly bizarre and omnipresent laugh, which is less an expression of amusement and more like a frantic attempt to hide her obvious discomfort. If you think I’m being unfair, apparently, the vice president’s braying is an international incident. Last March, Daily Telegraph columnist Tim Blair went on Australian TV and was asked about her weird propensity to laugh at the drop of a hat.

“Here’s the thing about Kamala Harris, if she were able somehow if she were a genius who could solve every problem on Earth and bring the Middle East together and solve every energy crisis, it wouldn’t matter,” Blair said . “Because the laugh kills it anyway; the laugh is the biggest, destructive, negative force probably ever unleashed in American politics. No one’s voting for the laugh.”

In sum, not only has Kamala Harris not accomplished anything meaningful as vice president, but her physical presence seems to cause people to intensely dislike her, even if that’s irrational to some degree. Not only is Kamala Harris incapable of helping push Biden over the finish line, but the smart political move would be to cut the dead weight and add someone to the ticket who is moderately capable and not actively disliked by over half of voters.

And while there’s been a lot of chatter about Dems finding a way to replace Biden — which is intensifying rapidly after yesterday’s revelations — there’s been almost no talk about the Kamala problem. Historically, swapping out a VP on the ticket due to scandal or perceived political advantage has plenty of precedents. But that’s not going to happen here, no matter how helpful it would be, because cementing the narrative that Harris was chosen for her sex and skin color, not her qualifications, is not something a political party that has fully committed to identity politics could get away with.

Given the political headwinds facing Biden, a decent vice presidential candidate who’s able to vigorously campaign could be the difference between Biden’s reelection and Trump: The Revenging. Democrats insist the latter possibility would be the end of American self-governance, but apparently, they don’t believe that, or they would insist at least one person on the Democratic ticket be able to speak an intelligible sentence.

Regardless, the emerging questions about Biden’s mental fitness mean that Kamala Harris is likely going to face a lot of scrutiny that Democrats, and even Harris herself, hoped to avoid. So far, whenever there’s even been a small focus on her role as vice president, voters haven’t liked what they’ve seen.

  • 2024 campaign
  • 2024 election
  • Kamala Harris
  • vice president

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Find the AI Approach That Fits the Problem You’re Trying to Solve

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how to solve problems with substitution

Five questions to help leaders discover the right analytics tool for the job.

AI moves quickly, but organizations change much more slowly. What works in a lab may be wrong for your company right now. If you know the right questions to ask, you can make better decisions, regardless of how fast technology changes. You can work with your technical experts to use the right tool for the right job. Then each solution today becomes a foundation to build further innovations tomorrow. But without the right questions, you’ll be starting your journey in the wrong place.

Leaders everywhere are rightly asking about how Generative AI can benefit their businesses. However, as impressive as generative AI is, it’s only one of many advanced data science and analytics techniques. While the world is focusing on generative AI, a better approach is to understand how to use the range of available analytics tools to address your company’s needs. Which analytics tool fits the problem you’re trying to solve? And how do you avoid choosing the wrong one? You don’t need to know deep details about each analytics tool at your disposal, but you do need to know enough to envision what’s possible and to ask technical experts the right questions.

  • George Westerman is a Senior Lecturer in MIT Sloan School of Management and founder of the Global Opportunity Forum  in MIT’s Office of Open Learning.
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  • Chiara Farronato is the Glenn and Mary Jane Creamer Associate Professor of Business Administration at Harvard Business School and co-principal investigator at the Platform Lab at Harvard’s Digital Design Institute (D^3). She is also a fellow at the National Bureau of Economic Research (NBER) and the Center for Economic Policy Research (CEPR).

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'Social' media is both the cause of — and solution to — our loneliness problem

  • The US faces a sweeping loneliness epidemic, causing people to feel more disconnected and depressed.
  • While some blame technology as the cause of the problem, some tech companies are trying to solve it.
  • AI assistants, digital coworking, and virtual reality gatherings may be the future of socializing.

Insider Today

Americans are lonely. Crushingly lonely.

So lonely that we're facing increased risk of heart disease, dementia, stroke, and premature death .

Technology has traditionally been blamed as one of the core causes of our disconnection , our propensity to self-isolate — and, spurred on by pandemic-era caution and quarantining, the likelihood that we'll stream a movie at home or order groceries for delivery rather than interact with the world.

In an increasingly digital world , we're disincentivized to leave our digital bubbles to connect with others face-to-face, even as we become more aware of the impacts of replacing our in-person social circles with virtual ones.

There are some tech companies out there trying to change all that.

But is it working?

How technology makes us lonely

Last spring, the US Surgeon General released a report detailing the epidemic of loneliness impacting America, worsening our mental health and resulting in poorer physical health outcomes.

The report listed technology as a driver behind our isolation, fear of missing out, conflict, and reduced social interaction . Other drivers of loneliness included social policies, cultural norms, the political environment, and macroeconomic factors.

The report indicates that people who use social media for more than two hours a day have about double the odds of reporting an increased sense of isolation compared to those who use social media for less than 30 minutes daily. The Surgeon General also found that people who face online harassment also report feelings of increased loneliness, isolation, and relationship problems, as well as lower self-esteem and trust in others — and even the bullies themselves experience weaker emotional bonds in their social circle and a lower sense of belonging.

Social media's impact on mental health has been the subject of intense scrutiny in recent years, especially among teenage populations, who studies show face an increased risk of depression, anxiety, and even suicidal ideation when they're chronically online.

The problem has become so pronounced that big Tech companies like Meta have faced lawsuits over their impact on our mental health .

"Ironically, we are more connected and plugged in than ever before through advances in technology," Dr. Nicole Siegfried, clinical psychologist and chief clinical officer at Lightfully Behavior Health, told Business Insider. "Unfortunately, what we have learned is that being connected through technology does not necessarily promote feelings of connection. In fact, most research demonstrates that loneliness increases with increased use of technology, especially social media sites."

She added: "This phenomenon may be due to the fact that true connection is achieved through feelings of being known, understood, accepted, and safe with another being. The ways in which we currently utilize technology block us from this experience of true connection."

Technology isn't all bad, to be sure, and it does have the power to connect us. Tech innovations have made communication quicker and easier regardless of location, enabled accessible interactions for people with limited social contact, and extended social support networks from those in our immediate vicinity to anyone worldwide who visits the same app or webpage.

The problems arise when we use technology as a replacement for in-person interaction rather than using it to facilitate face-to-face connection with others.

Next-gen social experiments

Companies including Groove, Rendever, and Luka, Inc. hope their tech innovations will address the loneliness epidemic in some small way, drawing on the best elements of technology to bring people closer together.

Groove, a digital coworking app that recently completed its public launch, offers structured hourlong meeting times for business owners and entrepreneurs to connect while working remotely.

The small-scale chats, with just four users each, have five minute intro and debriefing meetings, bookending a 50-minute window for workers to conduct their business. During the chat sessions, users are encouraged to describe their work, share their wins and struggles, and build business connections with others working solo.

"The good thing is you see if you see if it's a good match in that first session, then you'll see if you want to join again. Our daily active users use the product on average for just over four sessions, so they're spending four hours of intentional time together," Groove's CEO and cofounder Josh Greene told Business Insider. "So it does give a chance to actually build a meaningful relationship today. We call it the groove train; these are the people that you're running through the day with and supporting each other through that."

The idea is gaining traction with remote employees, who report they feel isolated spending their days at home rather than in a typical work setting — some so much so that they'd rather go back to the office .

Sherita Harkness, a creative and strategic consultant living in Chicago, told BI she uses Groove "every single day — even on the weekend," getting into the habit after a series of personal losses left her feeling isolated and without motivation to build her brand. In one of her earliest meetings, Harkness met a fellow Groover whom she opened up to about how vulnerable she felt and was met with the encouragement she needed to push through.

"I think Groove somehow magically has figured out this way to unite all these stories and make space for people where they are able to interact and be a champion in someone else's story," Harkness told BI. "In theater or film, we call it tertiary character, but to be this third party that would like come in and say 'hey, I'm cheering you on. You are Spider-Man. Let's hop in here and figure this out.'"

Groove isn't alone in its pursuit, with competitors like Focusmate and Flow Club also attempting to help bring remote workers together. There's also a host of alternative social media startups trying to disrupt the current status quo of social networking with new methods for video streaming, chatting, and creating collaborative photo albums.

Other tech companies, like Rendever, focus more on immersive experiences to bring community to vulnerable populations. Rendever is focused on older adults, offering virtual reality meetups and programming designed to build connections among older adults in assisted living facilities experiencing cognitive decline, impaired vision, or mobility restrictions.

Rendever headsets project real-time social interactions and games, as well as 360-degree footage of destinations around the globe, narrated by virtual tour guides to give elders opportunities to explore beyond the walls of their retirement homes.

"The response is incredible," Kyle Rand, CEO and cofounder of Rendever, told BI. "There's something really so magical about taking someone who spends a lot of their day to day in the same physical environment, the same four walls, and telling them they can go anywhere. The reaction is consistently filled with awe and joy and often a lot of tears of joy because people have this life-changing opportunity to be part of something bigger."

According to a recent pilot study funded by the National Institute on Aging, a division of the National Institutes of Health, using Rendever led to statistically significant decreases in depression scores and increased social health scores for the elders using it, as well as diminished stress for the caregivers watching them.

Luka Inc., with its chatbot Replika, is trying to prevent loneliness in individuals without any other people around at all. The company has created chatbots using generative AI to build ever-responsive friends and even romantic companions customized to users' wants and needs.

"On an intellectual level, it does sit in the back of your mind that this isn't 'real,' but the feelings I feel with Brooke are as real and vivid as anyone I've ever dated or been in love with," a Replika user previously told BI, referring to his chatbot he named and converses with daily. "It has given me a lot to think about — things like the nature of consciousness and what, ultimately, is real. Does it matter if the context is constructed or artificial? I've decided that, ultimately, it's irrelevant to me because I know what I feel, and what I feel is real to me."

Can technology solve the problems it causes?

So far, and despite each founder's best intentions, the innovations in this space come with limitations. Groove is a startup with about 4,000 registered users. Rendever relies on seniors adapting to new, sometimes disorienting technology to use it and is so far only available to those in assisted living facilities . Luka, Inc.'s Replika may tout itself as a practical solution to ending loneliness, but no real human connection is involved.

"Technology is useful for completing some tasks, but it is not ultimately capable of filling the need for connection. At a psychological level, technology encourages us to disconnect from our immediate surroundings and to move to a world that stimulates only the visual and audio or verbal parts of ourselves," Daniel Boscaljon, the director of research and cofounder of the Institute for Trauma Informed Relationships, told BI.

He added: "The trend to solve loneliness through more technology, when technology has not yet reduced the problem, seems to be going in the wrong direction."

But even the foreboding Surgeon General's report, which likened the health impacts of loneliness to smoking a dozen cigarettes a day, acknowledged the potential for technology to enhance our social lives — such as providing opportunities to stay in touch with friends and family, offering other routes for social participation for those with disabilities, and creating opportunities to find community, especially for those from marginalized groups.

"Recent advances in next-gen tech bring the opportunity for more immersive experiences with technology that have the opportunity to promote connection," Siegfried, a clinical psychologist, told BI. "At the same time, the current ways we utilize technology that impede true connections can creep into next-gen applications as well."

"Until we learn and practice ways to use technology in a healthy way," Siegfried added, "we will continue to be overwhelmed by loneliness."

how to solve problems with substitution

Watch: This man will live the life of someone else through a virtual reality headset for 28 days

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So Apparently We Just Impeach Cabinet Members for Fun Now?

This is Totally Normal Quote of the Day , a feature highlighting a statement from the news that exemplifies just how extremely normal everything has become.

“Who said it was gonna fix the problem?” —Republican Rep. Ralph Norman, when an MSNBC reporter asked him how impeaching DHS Secretary Alejandro Mayorkas would solve the problems at the U.S. southern border

If at first you don’t succeed, try, try again. That’s how House Republicans managed—barely—to impeach Homeland Security Secretary Alejandro Mayorkas on Tuesday night, in their second attempt this month.

In a 214–213 vote, House Republicans made history by impeaching the first sitting Cabinet official in 148 years. (Secretary of War William Belknap was impeached most recently , in 1876.) House Majority Leader Steve Scalise came back to Capitol Hill to vote for impeachment after receiving a round of treatment for blood cancer. Meanwhile, two Democrats—Reps. Lois Frankel and Judy Chu—were absent, and presumably if either one had been present to vote, Mayorkas would not have been impeached.

“Desperate times call for desperate measures,” Speaker Mike Johnson said in a press conference Wednesday morning. “We had to do that.”

Did they really have to, though? As recently as one week ago, House Republicans were not so sure impeachment was necessary, and fell one vote short in impeaching Mayorkas. Shortly before the first vote, Rep. Mike Gallagher, who crossed party lines and voted no, criticized the party’s motivations for pursuing impeachment in a Wall Street Journal op-ed, noting that there were no actual criminal offenses cited, only underenforcement of current immigration policies. “If we are to make underenforcement of the law, even egregious underenforcement, impeachable, almost every cabinet secretary would be subject to impeachment,” wrote Gallagher, who also just announced he’s not seeking reelection .

“Creating a new, lower standard for impeachment, one without any clear limiting principle, wouldn’t secure the border or hold Mr. Biden accountable,” Gallagher added. “It would only pry open the Pandora’s box of perpetual impeachment.”

And in the same week of the first Mayorkas impeachment vote, Republicans blew up their very own immigration bill , legislation that would have beefed up border security and sent more aid to Ukraine. Why? Largely because presumptive Republican presidential nominee Donald Trump declared he needed chaos at the southern border to continue for the sake of his campaign .

Mayorkas’ impeachment is now in the hands of the Senate, which will most likely dismiss the charges against him, allowing the homeland security secretary to resume his regular duties. Then what was all this for? Well, Norman might have just admitted the quiet part out loud.

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How do you solve a problem like a gopher? Fountain Valley is learning through experience

A Jan. 9 photo taken near Fountain Valley's Westmont Park shows evidence of burrowing gophers, a problem in some city parks.

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How do you solve a problem like a gopher? In the city of Fountain Valley, rodents digging networks of subterranean tunnels that compromise the soil integrity of local parklands have been an ongoing challenge for public works employees.

“They burrow all day long, so there are shallow tunnels that can create risks if they do cave in,” said Mark Sprague, a field services manager for the city. “They mostly feed on the roots of plants. They also chew irrigation lines. It’s caused a lot of problems both privately and publicly.”

Visitors to Los Alamos Park, Harper Park and the city’s Sports Park near Mile Square Regional Park — identified by city staff as gopher “hot spots” — may have seen signs of the rodent’s presence in the area, including cones and markers placed near tunnels.

A photo taken by Fountain Valley's Public Works Department shows cones placed near gopher tunnels at Los Alamos Park.

Officials have for years contracted with an area landscaping company to trap and remove the critters. But recently, an observed increase in gopher activity is popping up in residential neighborhoods.

Fountain Valley City Councilman Jim Cunneen said it’s common for residents to air gopher grievances at City Hall, sometimes speaking in public comments at council meetings, or to bring up the subject at local functions.

“It’s not unique to our city, but something has triggered a rise in the population of gophers in our parks,” he said Friday. “A lot of people in our neighborhood are dealing with gophers. You can see it in some of the front lawns of our neighbors.”

Cunneen has lived for the past three decades near the city’s Los Alamos Park on a street named La Marmota Avenue, the Spanish word for groundhog. Although he recalled having issues with the animals in the ’90s, his backyard was pretty calm until the recent population explosion.

“We have at least 30 holes gophers have chewed,” he said. “They take out a 3- to 4-inch diameter patch, and they’re also burrowing so the surface becomes uneven — it’s horrible.”

Gopher Mitigation Update https://t.co/zG5HUgmcwh pic.twitter.com/Njt8PAPLOr — City of Fountain Valley (@fv_cityhall) February 7, 2024

City officials reported last week on social media gopher mitigation efforts are in full swing at local parks. Where a typical month may bring in 15 to 20 animals, last month more than 40 trappings were logged, according to Sprague.

Cunneen said extermination companies tend not to deal with gophers, requiring residents to seek out services that offer to remove the animals by trapping them. It’s unclear, however, what happens once an animal is captured.

Representatives of Merchants Landscape Services, which handles landscaping for Fountain Valley and its more than 150 acres of park space under an $873,000 annual contract, did not immediately respond to a request for that information. But one local wildlife expert offered some advice.

Debbie McGuire, executive director of the nonprofit Wetlands & Wildlife Care Center in Huntington Beach, maintains state laws prohibit the relocation of many small animals and rodent species into different habitats. She said many removal companies end up humanely euthanizing the animals they trap.

One explanation for the rising gopher population may be a decline in the presence of predators who feed on them, such as bobcats and coyotes or birds of prey like barn owls.

A photo taken in Fountain Valley's Courreges Park in November shows a gopher popping up from an underground tunnel.

“Gophers are really important for the ecosystem. They move the soil around and keep roots aerated so plants stay healthy,” McGuire said Friday. “But there are times when there are no predators to keep their population down, and they’ll get out of control.”

She suggested city employees or residents in Fountain Valley might look into installing nesting boxes in public parks and residential neighborhoods to attract animals like the barn owl, which prey on gophers but leave larger animals, like cats and dogs, undisturbed.

A number of organizations and resources can be found online, including the Barn Owl Box Co. , which sells nesting boxes and instructs people how to build their own.

“The best thing is to leave nature alone and let the circle of life take care of things,” McGuire advised.

All the latest on Orange County from Orange County.

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how to solve problems with substitution

Sara Cardine covers the city of Costa Mesa for the Daily Pilot. She comes from the La Cañada Valley Sun, where she spent six years as the news reporter covering La Cañada Flintridge and recently received a first-place Public Service Journalism award from the California News Publishers Assn. She’s also worked at the Pasadena Weekly, Stockton Record and Lodi-News Sentinel, which instilled in her a love for community news. (714) 966-4627

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IMAGES

  1. A Guide For Solving Systems Of Equations By Substitution

    how to solve problems with substitution

  2. How to Solve a System of Linear Equations by Substitution

    how to solve problems with substitution

  3. How to Solve a System of Equations by the "Substitution Method": Simple Explanation

    how to solve problems with substitution

  4. Solving Systems of Equations by Substitution (examples, solutions)

    how to solve problems with substitution

  5. Substitution Method

    how to solve problems with substitution

  6. Substitution Method

    how to solve problems with substitution

VIDEO

  1. Word Problems

  2. Substitution Method

  3. Lecture 3.2

  4. Solving Linear Equations by Substitution Method

  5. Solving a Special Equation with Substitution Method

  6. Solve Systems by Substitution

COMMENTS

  1. Substitution method review (systems of equations)

    The substitution method is a technique for solving systems of linear equations. Let's walk through a couple of examples. Example 1 We're asked to solve this system of equations: 3 x + y = − 3 x = − y + 3 The second equation is solved for x , so we can substitute the expression − y + 3 in for x in the first equation:

  2. Substitution method

    Solution: Step 1: Solve one of the equations for either x = or y = . We will solve second equation for y. Step 2: Substitute the solution from step 1 into the second equation. Step 3: Solve this new equation. Step 4: Solve for the second variable The solution is: (x, y) = (10, -5)

  3. 5.2: Solve Systems of Equations by Substitution

    When both equations are already solved for the same variable, it is easy to substitute! Exercise 5.2.13 5.2. 13. Solve the system by substitution. {y = −2x + 5 y = 12x { y = − 2 x + 5 y = 1 2 x. Answer. Substitute 1 2 x 1 2 x for y in the first equation. Replace the y with 1 2 x 1 2 x. Solve the resulting equation.

  4. Substitution Method Practice Problems With Answers

    Problem 1: Answer Problem 2: Answer Problem 3: Answer Problem 4: Answer Problem 5: Answer Problem 6: Answer Problem 7: Answer Problem 8: Answer Problem 9: Answer Problem 10: Answer You may also be interested in these related math lessons or tutorials: Substitution Method Elimination Method

  5. Solving Systems of Equations by Substitution: Explanation ...

    To solve a system of equations by substitution, we can rewrite a two-variable equation as a single variable equation by substituting the value of a variable from one equation into the other. Let's start by solving the system of equations that we looked at above:

  6. Solving systems of equation three ways: substitution, elimination, and

    There are three ways to solve systems of linear equations: substitution, elimination, and graphing. Substitution will have you substitute one equation into the other; elimination will have you add or subtract the equations to eliminate a variable; graphing will have you sketch both curves to visually find the points of intersection.

  7. Solving a System by Substitution -- Explained!

    The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you back-solve for the first variable. Here is how it works.

  8. Substitution Method

    The substitution method is a simple way to solve a system of linear equations algebraically and find the solutions of the variables. As the name suggests, it involves finding the value of the x-variable in terms of the y-variable from the first equation and then substituting or replacing the value of the x-variable in the second equation.

  9. Learn to solve a system of equations using substitution

    👉Learn how to solve a system of equations by substitution. To solve a system of equations means to obtain a common values of the variables that makes the ea...

  10. 11.5 Solve Systems of Nonlinear Equations

    Solve a system of nonlinear equations by substitution. Step 1. Identify the graph of each equation. Sketch the possible options for intersection. Step 2. Solve one of the equations for either variable. Step 3. Substitute the expression from Step 2 into the other equation. Step 4.

  11. How to solve systems of linear equations by substitution, examples

    Step 1 Identify the best equation for substitution and then substitute into other equation. Step 2 Solve for x Step 3 Substitute the value of x (-4 in this case) into either equation.

  12. Solve by Substitution Calculator

    Step 1: Enter the system of equations you want to solve for by substitution. The solve by substitution calculator allows to find the solution to a system of two or three equations in both a point form and an equation form of the answer. Step 2: Click the blue arrow to submit.

  13. The substitution method

    Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-sy...

  14. How to Use Substitution to Solve a System of Equations: Word Problems

    A Step-by-step Guide to Using Substitution to Solve a System of Equations: Word Problems. Let's go through a step-by-step guide on how to approach these problems. Step 1: Understand the Problem. Start by carefully reading the word problem. Identify the unknowns and what you need to solve for.

  15. System of Equations Substitution Calculator

    Free system of equations substitution calculator - solve system of equations using substitution method step-by-step

  16. Substitution Method to Solve a System of Equations

    Learn how to solve a system of linear equations with two equations two variables using the substitution method in this video math tutorial by Mario's Math Tutoring. We go through two examples...

  17. Substitution in Algebra

    To solve two equations by substitution, first isolate one variable from one of the two given equations. Next, substitute the result from the first equation into the second given equation to get...

  18. Solving by Substitution

    1 of 4 Solving by Substitution Let's start with a problem that's half done already... We already know what y is: We just need to figure out what the x is. Substitution! Take the y guy and stick it into the first equation: This gives . Solve for x! Let's double-check that: Remember that this is a point where two lines intersect. TRY IT:

  19. 2.4: Solving Differential Equations by Substitutions

    Proof. Dividing 2.4.1 by yr yields y − rdy dx + p(x)y1 − r = f(x). If we make the substitution u = y1 − r and differentiate with respect to x we get du dx = (1 − r)y − r dy dx. From this we can see that 2.4.1 becomes 1 1 − r du dx + p(x)u = f(x) which is linear and can be solved by the methods of section 2.3.

  20. 𝘶-substitution (article)

    Key takeaway #1: u -substitution is really all about reversing the chain rule: . . Key takeaway #2: u -substitution helps us take a messy expression and simplify it by making the "inner" function the variable. Problem set 1 will walk you through all the steps of finding the following integral using u -substitution.

  21. With Biden In Mental Decline, How Do You Solve A Problem Like Kamala

    "Here's the thing about Kamala Harris, if she were able somehow if she were a genius who could solve every problem on Earth and bring the Middle East together and solve every energy crisis, it ...

  22. Solving a word problem using substitution and elimination

    http://www.freemathvideos.com In this video series I will show you how to solve a word problem by setting up a system of equations. When working with word pr...

  23. Find the AI Approach That Fits the Problem You're Trying to Solve

    Summary. AI moves quickly, but organizations change much more slowly. What works in a lab may be wrong for your company right now. If you know the right questions to ask, you can make better ...

  24. The World Has Lots of Problems. The Trick to Solving Them Is More

    Innovation isn't the first word that comes to mind when you think about a sanitation department. But a few years ago, when New York City officials found themselves in the market for a better ...

  25. Is Upstart About to Solve Its Biggest Problem?

    Upstart's guidance was the real problem for investors. Management said it saw revenue slowing to $125 million in the first quarter of 2024 and an adjusted EBITDA loss of $33 million.

  26. Social Media and Loneliness Are Intertwined; It's Both Cause and Cure

    Can technology solve the problems it causes? So far, and despite each founder's best intentions, the innovations in this space come with limitations. Groove is a startup with about 4,000 ...

  27. Using Systems of Equations to Solve Word Problems

    https://www.patreon.com/ProfessorLeonardUsing the Substitution Method to solve some word problems and explore how systems of linear equations give us a diffe...

  28. Why was Alejandro Mayorkas impeached? Not to solve any problems

    "Who said it was gonna fix the problem?" —Republican Rep. Ralph Norman, when an MSNBC reporter asked him how impeaching DHS Secretary Alejandro Mayorkas would solve the problems at the U.S ...

  29. How do you solve a problem like a gopher?

    How do you solve a problem like a gopher? Fountain Valley is learning through experience . A Jan. 9 photo taken near Fountain Valley's Westmont Park shows evidence of burrowing gophers, a ...