1. Warm Up
Solve each equation for x.
1. y = x + 3
2. y = 3x – 4
x = y – 3

2. Objective
Solve systems of linear equations in two variables by substitution.

3. Sometimes it is difficult to identify the exact solution to a system by graphing. In this case, you can use a method called substitution.
The goal when using substitution is to reduce the system to one equation that has only one variable. Then you can solve this equation by the methods taught in Chapter 2.

4. Solving Systems of Equations by Substitution
Step 2
Step 3
Step 4
Step 5
Solve for one variable in at least one equation, if necessary.
Step 1
Substitute the resulting expression into the other equation.
Solve that equation to get the value of the first variable.
Substitute that value into one of the original equations and solve.
Write the values from steps 3 and 4 as an ordered pair, (x, y), and check.

5. Example 1A: Solving a System of Linear Equations by Substitution
Solve the system by substitution.
y = 3x
y = x – 2
Step 1 y = 3x
Both equations are solved for y.
y = x – 2
Step 2 y = x – 2
3x = x – 2
Substitute 3x for y in the second equation.
Step 3
–x –x
2x = –2
2x = –2
x = –1
Solve for x. Subtract x from both sides and then divide by 2.

6.   Example 1A Continued Solve the system by substitution.
Write one of the original equations.
Step 4
y = 3x
y = 3(–1)
y = –3
Substitute –1 for x.
Write the solution as an ordered pair.
Step 5
(–1, –3)
Check Substitute (–1, –3) into both equations in the system.
y = 3x
–3 3(–1)
–3 –3 
y = x – 2
–3 –1 – 2
–3 –3 

7. Example 1B: Solving a System of Linear Equations by Substitution
Solve the system by substitution.
y = x + 1
4x + y = 6
The first equation is solved for y.
Step 1 y = x + 1
Step 2 4x + y = 6
4x + (x + 1) = 6
Substitute x + 1 for y in the second equation.
5x + 1 = 6
Simplify. Solve for x.
Step 3
–1 –1
5x = 5
x = 1
5x = 5
Subtract 1 from both sides.
Divide both sides by 5.

8.   Example1B Continued Solve the system by substitution.
Write one of the original equations.
Step 4
y = x + 1
y = 1 + 1
y = 2
Substitute 1 for x.
Write the solution as an ordered pair.
Step 5
(1, 2)
Check Substitute (1, 2) into both equations in the system.
y = x + 1
2 2 
4x + y = 6
4(1)
6 6 

9. Example 1C: Solving a System of Linear Equations by Substitution
Solve the system by substitution.
x + 2y = –1
x – y = 5
Step 1 x + 2y = –1
Solve the first equation for x by subtracting 2y from both sides.
−2y −2y
x = –2y – 1
Step 2 x – y = 5
(–2y – 1) – y = 5
Substitute –2y – 1 for x in the second equation.
–3y – 1 = 5
Simplify.

10. Example 1C Continued
Step 3
–3y – 1 = 5
Solve for y.
+1 +1
–3y = 6
Add 1 to both sides.
–3y = 6
–3 –3
y = –2
Divide both sides by –3.
Step 4
x – y = 5
Write one of the original equations.
x – (–2) = 5
x + 2 = 5
Substitute –2 for y.
–2 –2
x = 3
Subtract 2 from both sides.
Write the solution as an ordered pair.
Step 5 (3, –2)

11. Check It Out! Example 1a
Solve the system by substitution.
y = x + 3
y = 2x + 5
Step 1
y = x + 3
y = 2x + 5
Both equations are solved for y.
Step 2
2x + 5 = x + 3
y = x + 3
Substitute 2x + 5 for y in the first equation.
–x – 5 –x – 5
x = –2
Step 3
2x + 5 = x + 3
Solve for x. Subtract x and 5 from both sides.

12. Check It Out! Example 1a Continued
Solve the system by substitution.
Write one of the original equations.
Step 4
y = x + 3
y = –2 + 3
y = 1
Substitute –2 for x.
Step 5
(–2, 1)
Write the solution as an ordered pair.

13. Check It Out! Example 1b
Solve the system by substitution.
x = 2y – 4
x + 8y = 16
Step 1 x = 2y – 4
The first equation is solved for x.
(2y – 4) + 8y = 16
x + 8y = 16
Step 2
Substitute 2y – 4 for x in the second equation.
Step 3
10y – 4 = 16
Simplify. Then solve for y.
10y = 20
Add 4 to both sides.
10y =
Divide both sides by 10.
y = 2

14. Check It Out! Example 1b Continued
Solve the system by substitution.
Step 4
x + 8y = 16
Write one of the original equations.
x + 8(2) = 16
Substitute 2 for y.
x + 16 = 16
Simplify.
x = 0
– 16 –16
Subtract 16 from both sides.
Write the solution as an ordered pair.
Step 5 (0, 2)

15. Check It Out! Example 1c
Solve the system by substitution.
2x + y = –4
x + y = –7
Solve the second equation for x by subtracting y from each side.
Step 1 x + y = –7
– y – y
x = –y – 7
2(–y – 7) + y = –4
x = –y – 7
Step 2
Substitute –y – 7 for x in the first equation.
Distribute 2.
2(–y – 7) + y = –4
–2y – 14 + y = –4

16. Check It Out! Example 1c Continued
Solve the system by substitution.
Step 3
–2y – 14 + y = –4
Combine like terms.
–y – 14 = –4
–y = 10
Add 14 to each side.
y = –10
Step 4
x + y = –7
Write one of the original equations.
x + (–10) = –7
Substitute –10 for y.
x – 10 = – 7

17. Check It Out! Example 1c Continued
Solve the system by substitution.
Step 5
x – 10 = –7
Add 10 to both sides.
x = 3
Step 6
(3, –10)
Write the solution as an ordered pair.

18. Sometimes you substitute an expression for a variable that has a coefficient. When solving for the second variable in this situation, you can use the Distributive Property.

19. When you solve one equation for a variable, you must substitute the value or expression into the other original equation, not the one that had just been solved.
Caution

20. Example 2: Using the Distributive Property
y + 6x = 11
Solve by substitution.
3x + 2y = –5
Solve the first equation for y by subtracting 6x from each side.
Step 1 y + 6x = 11
– 6x – 6x
y = –6x + 11
3x + 2(–6x + 11) = –5
3x + 2y = –5
Step 2
Substitute –6x + 11 for y in the second equation.
Distribute 2 to the expression in parentheses.
3x + 2(–6x + 11) = –5

21. Example 2 Continued
y + 6x = 11
Solve by substitution.
3x + 2y = –5
Step 3
3x + 2(–6x) + 2(11) = –5
Simplify. Solve for x.
3x – 12x + 22 = –5
–9x + 22 = –5
–9x = –27
– 22 –22
Subtract 22 from both sides.
–9x = –27
– –9
Divide both sides by –9.
x = 3

22. Example 2 Continued
y + 6x = 11
Solve by substitution.
3x + 2y = –5
Write one of the original equations.
Step 4
y + 6x = 11
y + 6(3) = 11
Substitute 3 for x.
y + 18 = 11
Simplify.
–18 –18
y = –7
Subtract 18 from each side.
Step 5
(3, –7)
Write the solution as an ordered pair.

23. Check It Out! Example 2
–2x + y = 8
Solve by substitution.
3x + 2y = 9
Step 1 –2x + y = 8
Solve the first equation for y by adding 2x to each side.
+ 2x x
y = 2x + 8
3x + 2(2x + 8) = 9
3x + 2y = 9
Step 2
Substitute 2x + 8 for y in the second equation.
Distribute 2 to the expression in parentheses.
3x + 2(2x + 8) = 9

24. Check It Out! Example 2 Continued
–2x + y = 8
Solve by substitution.
3x + 2y = 9
Step 3
3x + 2(2x) + 2(8) = 9
Simplify. Solve for x.
3x + 4x = 9
7x = 9
7x = –7
–16 –16
Subtract 16 from both sides.
7x = –7
Divide both sides by 7.
x = –1

25. Check It Out! Example 2 Continued
–2x + y = 8
Solve by substitution.
3x + 2y = 9
Write one of the original equations.
Step 4
–2x + y = 8
–2(–1) + y = 8
Substitute –1 for x.
y + 2 = 8
Simplify.
–2 –2
y = 6
Subtract 2 from each side.
Step 5
(–1, 6)
Write the solution as an ordered pair.

26. Example 3: Consumer Economics Application
Jenna is deciding between two cell-phone plans. The first plan has a $50 sign-up fee and costs $20 per month. The second plan has a $30 sign-up fee and costs $25 per month. After how many months will the total costs be the same? What will the costs be? If Jenna has to sign a one-year contract, which plan will be cheaper? Explain.
Write an equation for each option. Let t represent the total amount paid and m represent the number of months.

27. Example 3 Continued
Total paid
sign-up fee
payment
amount
for each
month. is
plus
Option 1 t =
$50 +
$20 m
Option 2 t =
$30 +
$25 m
Step 1 t = m
t = m
Both equations are solved for t.
Step 2
m = m
Substitute m for t in the second equation.

28. Example 3 Continued
Step 3
m = m
Solve for m. Subtract 20m from both sides.
–20m – 20m
= m
Subtract 30 from both sides.
– –30
= m
Divide both sides by 5.
m = 4
20 = 5m
Step 4
t = m
Write one of the original equations.
t = (4)
Substitute 4 for m.
t =
t = 130
Simplify.

29. Example 3 Continued
Write the solution as an ordered pair.
Step 5
(4, 130)
In 4 months, the total cost for each option would be the same $130.
If Jenna has to sign a one-year contract, which plan will be cheaper? Explain.
Option 1: t = (12) = 290
Option 2: t = (12) = 330
Jenna should choose the first plan because it costs $290 for the year and the second plan costs $330.

30. Check It Out! Example 3
One cable television provider has a $60 setup fee and charges $80 per month, and the second has a $160 equipment fee and charges $70 per month.
a. In how many months will the cost be the same? What will that cost be.
Write an equation for each option. Let t represent the total amount paid and m represent the number of months.

31. Check It Out! Example 3 Continued
Total paid
payment
amount
for each
month. is
fee
plus
Option 1 t =
$60 +
$80 m
Option 2 t =
$160 +
$70 m
Step 1 t = m
t = m
Both equations are solved for t.
Step 2
m = m
Substitute m for t in the second equation.

32. Check It Out! Example 3 Continued
Step 3
m = m
Solve for m. Subtract 70m from both sides.
–70m –70m
m = 160
Subtract 60 from both sides.
– –60
10m = 100
Divide both sides by 10.
m = 10
Step 4
t = m
Write one of the original equations.
t = (10)
Substitute 10 for m.
t =
t = 860
Simplify.

33. Check It Out! Example 3 Continued
Step 5
(10, 860)
Write the solution as an ordered pair.
In 10 months, the total cost for each option would be the same, $860.
b. If you plan to move in 6 months, which is the cheaper option? Explain.
Option 1: t = (6) = 540
Option 2: t = (6) = 580
The first option is cheaper for the first six months.