1. Solving Systems by Substitution
5-2
Solving Systems by Substitution
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

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

3. 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.

4.   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 

5. 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.

6.   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 

7. 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.

8. 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)

9. 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.

10. 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.

11. 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

12. 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)

13. 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

14. 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

15. 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.

16. 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

17. 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

18. 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.

19. 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

20. 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

21. 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.

22. 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.

23. 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

29. 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.

30. Lesson Quiz: Part I
Solve each system by substitution. 1. 2. 3.
y = 2x
(–2, –4)
x = 6y – 11
(1, 2)
3x – 2y = –1
–3x + y = –1
x – y = 4

31. Lesson Quiz: Part II
4. Plumber A charges $60 an hour. Plumber B charges $40 to visit your home plus $55 for each hour. For how many hours will the total cost for each plumber be the same? How much will that cost be? If a customer thinks they will need a plumber for 5 hours, which plumber should the customer hire? Explain.
8 hours; $480; plumber A: plumber A is cheaper for less than 8 hours.