5.2 Solving Systems of Equations
by the Substitution Method
Solving Using Substitution
1. Solve either eqn. for x or y (may be done
already).
2. Substitute the expression for the variable
obtained in step 1 into the other eqn. and solve
it.
3. Substitute the value for the variable from step 2
into any eqn. in the system that contains both
variables and solve for the other variable.
4. Check soln. in BOTH eqns., if necessary.
Ex. Solve by the substitution method: x + y = 11
y = 2x – 1
1. y = 2x – 1
2.
x + y = 11
x + (2x – 1) = 11
x + 2x – 1 = 11
3x – 1 = 11
3x – 1 + 1 = 11 + 1
3x = 12
3x = 12
3
3
x=4
3. y = 2x – 1
y = 2(4) – 1 sub 4 for x
y=8–1
y=7
Soln: {(4, 7)}
4. Check:
x + y = 11
4 + 7 = 11
11 = 11
y = 2x – 1
7 = 2(4) – 1
7=8–1
7=7
Ex. Solve by the substitution method: 2x – 2y = 2
x – 5y = -7
1.
x – 5y = -7
x – 5y + 5y = -7 + 5y
x = 5y – 7
2.
2x – 2y = 2
2(5y – 7) – 2y = 2
10y – 14 – 2y = 2
8y – 14 = 2
8y + 14 – 14 = 2 + 14
8y = 16
8y = 16
8
8
y=2
3. x = 5y – 7
x = 5(2) – 7 sub 2 for y
x = 10 – 7
x=3
Soln: {(3, 2)}
4. Check:
2x – 2y = 2 x – 5y = -7
2(3) – 2(2) = 2 3 – 5(2)= -7
6 – 4 = 2 3 – 10 = -7
2=2
-7 = -7
Ex. Solve by the substitution method:
x
y
1


6
2
3
x + 2y = -3
First: eliminate fractions by
mult. by LCD
x
y
1


6
2
3
 x
 y
1
6   6   6 
6
2
3
x  3y  2
1. x – 3y = 2
x – 3y + 3y = 2 + 3y
x = 3y + 2
2.
x + 2y = -3
3y + 2 + 2y = -3
5y + 2 = -3
5y + 2 – 2 = -3 – 2
5y = -5
5y = -5
5
5
y = -1
3. x = 3y + 2
x = 3(-1) + 2 sub -1 for y
x = -3 + 2
x = -1
Soln: {(-1, -1)}
Ex. Solve by the substitution method:
1.
-x + y = -2
-x + y + x = -2 + x
y=x–2
2.
-4x + 4y = -8
-4x + 4(x – 2) = -8
-4x + 4x – 8 = -8
-8 = -8
-4x + 4y = -8
-x + y = -2
No variables remain and a
TRUE stmt. 
Lines coincide, so there are
infinitely many solns.
Soln: {(x, y)| -x + y = -2} or
{(x, y)| -4x + 4y = -8}
Ex. Solve by the substitution method:
6x + 2y = 7
y = 2 – 3x
1. y = 2 – 3x
2.
6x + 2y = 7
6x + 2(2 – 3x) = 7
6x + 4 – 6x = 7
4=7
No variables remain and a
FALSE stmt. 
lines are parallel, so there is
NO SOLN. (empty set)
no soln. or ø
Groups
Page 307 – 308: 33, 41, 45, 49