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The Substitution Method

A way to solve systems of linear equations in 2 variables

Video on Solving by Substitution

The Substitution Method
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 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
example 3
Now we have 1 equation and 1 unknown, we can solve this problem as the work below shows.
s
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. example 3
Step 2) Solve for x s
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

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



Solve the system below using substitution $$ y = x+1 \\ y = 2x +2 $$
Answer
Practice Problem four solution of system of equations
The solution of this system is the point of intersection : (-3,-4).
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
Answer


Problem 3)

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


Problem 4)

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

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


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


Problem 7)

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

Interactive System of Linear Equations

(Applet on its own page)
Slope Intercept Form
Point Slope Form
Standard Form
Slope?
Show Rise and Run ?
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