1. 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 Review Example 2
Substitution Example 1
Let's re-examine system pictured on the left.
y=2x+1 and y=4x−1
Step 1) We are going to use substitution like we did in review example
2 above

2. 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+1y=2⋅1+1=2+1=3 or you can substitute x =1 into the other
equationy=4x−1y=4⋅1−1=4−1=3solution = (1,3)
Substitution Example 2
What is the solution of the system of equations:
y=2x+12y=3x−2
Step 1) Identify the best
equation for substitution
and then substitute into
other equation.

3. Step 2) Solve for x
Step 3) Substitute the
value of x (-4 in this case)
into either equation
y=2x+1y=2⋅−4+1=−8+1=−72y=3x−22y=3⋅−4−2or
you can use the other
equation2y=−12−22y=−1412⋅2y=12⋅−14y=−7
The solution of this system is (-4,-7)
Show Graph
Substitution Practice Problems
Solve the system below using substitution y=x+1y=2x+2
Answer
Practice Problems
Problem 1)
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

4. determine y coordinate of solution
y=x−5y=−2−5=−7 The solution is the point (-2, -7)
problem 2) Use the substitution method to solve the system:
Line 1: y = 5x – 1
Line 2: 2y= 3x + 12

5. This system of lines has a solution at the point (2, 9).
Problem 3)
Use substitution to solve the system:
Line 1: y = 3x + 1
Line 2: 4y = 12x + 4

6. This system has an infinite number of solutions. because 12x +4 = 12x is always true for all
values of x.
Problem 4)
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.
Problem 5)
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).

7. Problem 6)
Use substitution to solve the system:
Line 1: y = 3x + 1
Line 2: 4y = 12x + 3
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 7)
Solve the system using substitution.
Line 1: y= x +5
Line 2: y= 2x +2

8. The solution of this system is the point of intersection : (3,8).
Interactive System of Linear Equations
(Applet on its own page)
Slope Intercept Form
Point Slope Form
Standard Form
Slope?
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