Solving Systems of Linear Equations by Substitution
Graphing is a useful tool for solving systems of equations, but it can sometimes be timeconsuming. A quicker way to solve systems is to isolate one variable in one equation, and substitute the resulting expression for that variable in the other equation. Observe:
Example 1: Solve the following system, using substitution:
The easiest variable to isolate is
y
in the first equation, because it has no coefficient:
y = 13  5x
In the second equation, substitute for
y
its equivalent expression:
3x = 15  3(13  5x)
Solve the equation:
3x = 15  39 + 15x
3x = 15x  24
12x =  24
x = 2
Now substitute this
x
value into the "isolation equation" to find
y
:
y = 13  5x = 13  5(2) = 13  10 = 3
Thus, the solution to the system is
(2, 3)
. It is useful to check this solution in both equations.
Note: Although we chose
y
in the first equation in the previous example, isolating any variable in any equation will yield the same solution.
Example 2: Solve the following system, using substitution:
2x + 4y

= 
36 

10y  5x

= 
0 

It is easier to work with the second equation, because there is no constant term:
5x = 10y
x = 2y
In the first equation, substitute for
x
its equivalent expression:
2(2y) + 4y = 36
Solve the equation:
4y + 4y = 36
8y = 36
y = 4.5
Plug this
y
value into the isolation equation to find
x
:
x = 2y = 2(4.5) = 9
Thus, the solution to the system is (9, 4.5).
Example 3: Solve the following system, using substitution:
2x  4y

= 
12 

3x

= 
21 + 6y


It is easiest to isolate
x
in the second equation, since the
x
term already stands alone:
x =
x = 7 + 2y
In the first equation, substitute for
x
its equivalent expression:
2(7 + 2y)  4y = 12
Solve the equation:
14 + 4y  4y = 12
14 = 12
Since
14≠12
, the system of equations has no solution. It is inconsistent (and independent). The two equations describe two parallel lines.
Example 4: Solve the following system, using substitution:
10x

= 
4y  68 

2y  5x

= 
34 

Either equation can be used to isolate the variable. We will isolate
y
in the second equation:
2y = 5x + 34
y =
y =
x + 17
In the first equation, substitute for
y
its equivalent expression:
10x = 4(
x + 17)  68
10x = 10x + 68  68
10x = 10x
0 = 0
Since
0 = 0
for any value of
x
, the system of equations has infinite solutions. Every ordered pair (
x, y
) which satisfies
y =
x + 17
(the isolation equation) is a solution to the system. The system is dependent (and consistent). The two equations describe the same line
y =
x + 17
.