The substitution method is a technique for solving systems of linear equations. It involves solving one equation for one variable in terms of the other and substituting this expression into the second equation. The key steps are: 1) solve one equation for one variable, 2) substitute this expression into the other equation, 3) solve the resulting equation for the remaining variable, 4) check the solution by substituting back into the original equations. Examples demonstrate both consistent and inconsistent systems of equations solved using this method.

1. 7.2
Solving Systems of Linear
Equations by Substitution

2. The Substitution Method
Another method (beside getting lucky with
trial and error or graphing the equations) that
can be used to solve systems of equations is
called the substitution method.
You solve one equation for one of the
variables, then substitute the new form of the
equation into the other equation for the solved
variable.
Martin-Gay, Developmental Mathematics 2

3. The Substitution Method
Solving a System of Linear Equations by the
Substitution Method
1) Solve one of the equations for y.
2) Substitute the value of y from step 1 into the
other equation.
3) Solve for x.
4) Substitute the value of x found in step 3 into
either equation containing both variables.
5) Check your answer.
Martin-Gay, Developmental Mathematics 3

4. The Substitution Method
Example
Solve the following system using the substitution method.
3x – y = 6 and – 4x + 2y = –8
Solving the first equation for y,
3x – y = 6
–y = –3x + 6 (subtract 3x from both sides)
y = 3x – 6 (multiply both sides by – 1)
Substitute this value for y in the second equation.
–4x + 2y = –8
–4x + 2(3x – 6) = –8 (replace y with result from first equation)
–4x + 6x – 12 = –8 (use the distributive property)
2x – 12 = –8 (simplify the left side)
2x = 4 (add 12 to both sides)
x=2 (divide both sides by 2) Continued.
Martin-Gay, Developmental Mathematics 4

5. The Substitution Method
Example continued
Substitute x = 2 into the first equation solved for y.
y = 3x – 6 = 3(2) – 6 = 6 – 6 = 0
Our computations have produced the point (2, 0).
Check the point in the original equations.
First equation,
3x – y = 6
3(2) – 0 = 6 true
Second equation,
–4x + 2y = –8
–4(2) + 2(0) = –8 true
The solution of the system is (2, 0).
Martin-Gay, Developmental Mathematics 5

6. The Substitution Method
Example
Solve the following system of equations using the
substitution method.
y = 2x – 5 and 8x – 4y = 20
Since the first equation is already solved for y, substitute
this value into the second equation.
8x – 4y = 20
8x – 4(2x – 5) = 20 (replace y with result from first
equation)
8x – 8x + 20 = 20 (use distributive property)
Continued.
20 = 20 (simplify left side)
Martin-Gay, Developmental Mathematics 6

7. The Substitution Method
Example continued
When you get a result, like the one on the previous
slide, that is obviously true for any value of the
replacements for the variables, this indicates that the
two equations actually represent the same line.
There are an infinite number of solutions for this
system. Any solution of one equation would
automatically be a solution of the other equation.
This represents a consistent system and the linear
equations are dependent equations.
Martin-Gay, Developmental Mathematics 7

8. The Substitution Method
Example
Solve the following system of equations using the substitution
method.
3x – y = 4 and 6x – 2y = 4
Solve the first equation for y.
3x – y = 4
–y = –3x + 4 (subtract 3x from both sides)
y = 3x – 4 (multiply both sides by –1)
Substitute this value for y into the second equation.
6x – 2y = 4
6x – 2(3x – 4) = 4 (replace y with the result from the first equation)
6x – 6x + 8 = 4 (use distributive property)
8=4 (simplify the left side) Continued.
Martin-Gay, Developmental Mathematics 8

9. The Substitution Method
Example continued
When you get a result, like the one on the previous
slide, that is never true for any value of the
replacements for the variables, this indicates that the
two equations actually are parallel and never
intersect.
There is no solution to this system.
This represents an inconsistent system, even though
the linear equations are independent.
Martin-Gay, Developmental Mathematics 9

10. The Substitution Method
Example continued
When you get a result, like the one on the previous
slide, that is never true for any value of the
replacements for the variables, this indicates that the
two equations actually are parallel and never
intersect.
There is no solution to this system.
This represents an inconsistent system, even though
the linear equations are independent.
Martin-Gay, Developmental Mathematics 9