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# The Substitution Method

## Video on Solving by Substitution

The Substitution Method
First, let's review how the substitution property works in general
 Review Example 1 Review Example 2

### Substitution Example 1

 Let's re-examine system pictured on the left. $\color{red}{y} = 2x + 1 \text{ and } \color{red}{y} = 4x -1$ Step 1) We are going to use substitution like we did in review example 2 above
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. $$y = 2x + 1 \\ y = 2\cdot \color{red}{1} + 1 = 2 + 1 =3 \\ \text{ or you can substitute x =1 into the other equation} \\ y = 4x -1 \\ y = 4\cdot \color{red}{1}- 1 = 4 - 1 = 3 \\ \text{solution = }( 1,3)$$

### Substitution Example 2

What is the solution of the system of equations: $y = 2x + 1 \\ 2y = 3x - 2$
 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 $$y = 2x + 1 \\ y = 2\cdot \color{Red}{-4} + 1 = -8 + 1 = -7 \\ 2y = 3x - 2\\ 2y = 3\cdot-4 -2 \\ \text{or you can use the other equation} \\ 2y = -12 -2 \\ 2y = -14 \frac{1}{2}\cdot2y =\frac{1}{2}\cdot-14 \\ y = -7$$
The solution of this system is (-4,-7)
 Show Graph
Substitution Practice Problems

Solve the system below using substitution $$y = x+1 \\ y = 2x +2$$
Practice Problems

Problem 1)

Use substitution to solve the following system of linear equations:
• Line 1: y=3x – 1
• Line 2: y= x – 5
 Step 1

problem 2) Use the substitution method to solve the system:
• Line 1: y = 5x – 1
• Line 2: 2y= 3x + 12

Problem 3)

Use substitution to solve the system:
• Line 1: y = 3x + 1
• Line 2: 4y = 12x + 4

Problem 4)

Solve the system of linear equations by substitution
• Line 1: y= x + 2
• Line 2: y= x + 8

Problem 5)
Use the substitution method to solve the system:
• Line 1: y= x + 1
• Line 2: 2y= 3x

Problem 6)
Use substitution to solve the system:
• Line 1: y = 3x + 1
• Line 2: 4y = 12x + 3

Problem 7)

Solve the system using substitution.
• Line 1: y= x +5
• Line 2: y= 2x +2