# ##\$ Substitution Method

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## How to Use the Substitution Method to Solve Systems of Equations

1. Isolate a variable in one of the equations.
2. Substitute the isolated variable into the other equation.
3. Solve for the non-substituted variable.
4. Solve for the substituted variable by plugging the answer from Step 3 into either one of the original equations.

## Examples

Use the substitution method to solve this system of equations:

$y=2x-2$

$y=5x-23$

Which variable is easiest to isolate and substitute?

$y=2x-2$

$y=5x-23$

In this system of equations, the $$y$$ is already isolated in both equations.

What is the result when you substitute $$2x-2$$ for $$y$$?

$y=2x-2$

$y=5x-23$

The first equation tells me that $$y$$ is equal to $$\yellow 2x-2$$.

So, I will substitute $$\yellow 2x-2$$ everywhere I see a $$y$$ in the second equation.

$({\yellow 2x-2})=5x-23$

What does x equal?

To find the x-coordinate of the solution, I will solve the substituted equation until $$\yellow x$$ is by itself.

$\yellow (2x-2)=5x-23$

First, I will distribute the invisible coefficient of 1 to the terms inside the parentheses.

$\yellow 2x-2=5x-23$

Then I will subtract $$\yellow 2x$$ from both sides of the equation to combine like terms.

$\yellow -2=3x-23$

Next, I will add $$\yellow 23$$ to both sides to undo the subtraction.

$\yellow 21=3x$

Then I will divide both sides of the equation by $$\yellow 3$$ to undo the multiplication.

$\yellow 7=x$

The x-coordinate of my solution is $$\yellow 7$$.

What does y equal?

Now that I know that $$\yellow x=7$$, I can plug $$\yellow 7$$ into either one of the original equations to find the value of $$y$$.

$y=2x-2$

$y=5x-23$

I think it looks slightly easier to plug $$\yellow x=7$$ into the first equation, so I will do that.

$y=2({\yellow 7})-2$

I will simplify the right side of the equation by multiplying $$2$$ and $$\yellow 7$$.

$y=14-2$

Then I will subtract $$2$$ from $$14$$.

$y=12$

The y-coordinate of my solution is $$\yellow 12$$.

$$\yellow (7, 12)$$ is the solution to this system of equations.

$y=2x-2$

$y=5x-23$

I can check my answer by plugging $$\yellow x=7$$ and $$\yellow y=12$$ into both equations.

First Equation

${\yellow (12)}=2{\yellow(7)}-2$

$$2$$ times $$7$$ is $$14$$.

And $$14$$ minus $$2$$ does equal $$12$$.

So, $$\yellow (7, 12)$$ is a solution to the first equation.

Second Equation

${\yellow (12)}=5{\yellow(7)}-23$

$$5$$ times $$7$$ is $$35$$.

And $$35$$ minus $$23$$ does equal $$12$$.

So, $$\yellow (7, 12)$$ is a solution to the second equation.

Use the substitution method to solve this system of equations:

$x=-3y+19$

$-2x+y=18$

Which variable is easiest to isolate and substitute?

$x=-3y+19$

$-2x+y=18$

In this system of equations, the $$x$$ is already isolated in the first equation.

What is the result when you substitute $$-3y+19$$ for $$x$$?

$x=-3y+19$

$-2x+y=18$

The first equation tells me that $$x$$ is equal to $$\green -3y+19$$.

So, I will substitute $$\green -3y+19$$ everywhere I see a $$x$$ in the second equation.

$-2({\green -3y+19})+y=18$

What does y equal?

To find the y-coordinate of the solution, I will solve the substituted equation until $$\green y$$ is by itself.

$\green -2(-3y+19)+y=18$

First, I will distribute the $$\green -2$$ coefficient to the terms inside the parentheses.

$\green 6y-38+y=18$

Then I will add the $$\green 6y$$ and $$\green y$$ on the left side of the equation to combine like terms.

$\green 7y-38=18$

Next, I will add $$\green 38$$ to both sides of the equation to undo the subtraction.

$\green 7y=56$

Then I will divide both sides of the equation by $$\green 7$$ to undo the multiplication.

$\green y=8$

The y-coordinate of my solution is $$\green 8$$.

What does x equal?

Now that I know that $$\green y=8$$, I can plug $$\green 8$$ into either one of the original equations to find the value of $$x$$.

$x=-3y+19$

$-2x+y=18$

I think it looks easier to plug $$\green y=8$$ into the first equation, so I will do that.

$x=-3({\green 8})+19$

I will simplify the right side of the equation by multiplying $$-3$$ and $$\green 8$$.

$x=-24+19$

Then I will add $$-24$$ and $$19$$.

$x=-5$

The x-coordinate of my solution is $$\green -5$$.

$$\green (-5,8)$$ is the solution to this system of equations.

$x=-3y+19$

$-2x+y=18$

I can check my answer by plugging $$\green x=-5$$ and $$\green y=8$$ into both equations.

First Equation

${\green (-5)}=-3{\green (8)}+19$

$$-3$$ times $$8$$ is $$-24$$.

And $$-24$$ plus $$19$$ does equal $$-5$$.

So, $$\green (-5, 8)$$ is a solution to the first equation.

Second Equation

$-2{\green (-5)}+{\green (8)}=18$

$$-2$$ times $$-5$$ is $$10$$.

And $$10$$ plus $$8$$ does equal $$18$$.

So, $$\green (-5, 8)$$ is a solution to the second equation.

Use the substitution method to solve this system of equations:

$7x-2y=-19$

$2x+y=15$

Which variable is easiest to isolate and substitute?

$7x-2y=-19$

$2x+y=15$

I think that it would be easiest to isolate the $$y$$ in the second equation.

$2x+y=15$

To get $$y$$ by itself, I need to subtract $$2x$$ from both sides of the equation.

$y=-2x+15$

What is the result when you substitute $$-2x+15$$ for $$y$$?

$7x-2y=-19$

$y=-2x+15$

The rearranged version of the second equation tells me that $$y$$ is equal to $$\purple -2x+15$$.

So, I will substitute $$\purple -2x+15$$ everywhere I see a $$y$$ in the first equation.

$7x-2({\purple -2x+15})=-19$

What does x equal?

To find the x-coordinate of the solution, I will solve the substituted equation until $$\purple x$$ is by itself.

$\purple 7x-2(-2x+15)=-19$

First, I will distribute the $$\purple -2$$ coefficient to the terms inside the parentheses.

$\purple 7x+4x-30=-19$

Then I will add the $$\purple 7x$$ and $$\purple 4x$$ on the left side of the equation to combine like terms.

$\purple 11x-30=-19$

Next, I will add $$\purple 30$$ to both sides of the equation to undo the subtraction.

$\purple 11x=11$

Then I will divide both sides of the equation by $$\purple 11$$ to undo the multiplication.

$\purple x=1$

The x-coordinate of my solution is $$\purple 1$$.

What does y equal?

Now that I know that $$\purple x=1$$, I can plug $$\purple 1$$ into either one of the original equations to find the value of $$y$$.

$7x-2y=-19$

$2x+y=15$

I think it looks easiest to plug $$\purple x=1$$ into the second equation, so I will do that.

$2({\purple 1})+y=15$

I will simplify the right side of the equation by multiplying $$2$$ and $$\purple 1$$.

$2+y=15$

Then I will subtract $$2$$ from both sides to undo the addition.

$y=13$

The y-coordinate of my solution is $$\purple 13$$.

$$\purple (1, 13)$$ is the solution to this system of equations.

$7x-2y=-19$

$2x+y=15$

I can check my answer by plugging $$\purple x=1$$ and $$\purple y=13$$ into both equations.

First Equation

$7{\purple (1)}-2{\purple (13)}=-19$

$$7$$ times $$1$$ is $$7$$.

And $$2$$ times $$13$$ is $$26$$.

$$7$$ minus$$26$$ does equal $$-19$$.

So, $$\purple (1, 13)$$ is a solution to the first equation.

Second Equation

$2{\purple (1)}+{\purple (13)}=15$

$$2$$ times $$1$$ is $$2$$.

And $$2$$ plus $$13$$ does equal $$15$$.

So, $$\purple (1, 13)$$ is a solution to the second equation.