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 up above.

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

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

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

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

The solution of this system is (-4,-7)

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

Substitution Practice Problems

Problem 1

Solve the system below using substitution

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

The solution of this system is the point of intersection : (-3,-4).

Practice Problem four solution of system of equations
Problem 2

Use substitution to solve the following system of linear equations:

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

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

Step 2

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)

Problem 3

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).

Problem 4

Use substitution to solve the system:

  • Line 1: y = 3x + 1
  • Line 2: 4y = 12x + 4
Infinite Solutions of system of equations by substitution method

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

Problem 5

Solve the system of linear equations by substitution

  • Line 1: y= x + 2
  • Line 2: y= x + 8
Practice Problem Nine solution of system of equations

This system of linear equation has no solution.

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

Problem 6

Use the substitution method to solve the system:

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

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

Problem 7

Use substitution to solve the system:

  • Line 1: y = 3x + 1
  • 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

This system has an no solutions.
Problem 8

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).

back to Systems of Equations Next to Solve by Elmination