1. Solve the equation.
1. 6a – 3 + 2a = 13
ANSWER
a = 2
2. 4(n + 2) – n = 11
ANSWER
n = 1

2. 3. You burned 8 calories per minute on a treadmill and 10 calories per minute on an elliptical trainer for a total of 560 calories in 60 minutes. How many minutes did you spend on each machine?
ANSWER
treadmill: 20 min,
elliptical trainer: 40 min

3. Use the substitution method
EXAMPLE 1
Use the substitution method
Solve the linear system:
y = 3x + 2
Equation 1
x + 2y = 11
Equation 2
SOLUTION
STEP 1
Solve for y. Equation 1 is already solved for y.

4. Use the substitution method
EXAMPLE 1
Use the substitution method
STEP 2
Substitute 3x + 2 for y in Equation 2 and solve for x.
x + 2y = 11
Write Equation 2.
x + 2(3x + 2) = 11
Substitute 3x + 2 for y.
7x + 4 = 11
Simplify.
7x = 7
Subtract 4 from each side.
x = 1
Divide each side by 7.

5. EXAMPLE 1
Use the substitution method
STEP 3
Substitute 1 for x in the original Equation 1 to find the value of y.
y = 3x + 2 = 3(1) + 2 = = 5
ANSWER
The solution is (1, 5).

6. Use the substitution method
EXAMPLE 1
GUIDED PRACTICE
Use the substitution method
CHECK
Substitute 1 for x and 5 for y in each of the original
equations.
y = 3x + 2
x + 2y = 11
5 = 3(1) + 2 ?
1 + 2 (5) = 11 ?
5 = 5
11 = 11

7. Use the substitution method
EXAMPLE 2
Use the substitution method
Solve the linear system:
x – 2y = –6
Equation 1
4x + 6y = 4
Equation 2
SOLUTION
STEP 1
Solve Equation 1 for x.
x – 2y = –6
Write original Equation 1.
x = 2y – 6
Revised Equation 1

8. Use the substitution method
EXAMPLE 2
Use the substitution method
STEP 2
Substitute 2y – 6 for x in Equation 2 and solve for y.
4x + 6y = 4
Write Equation 2.
4(2y – 6) + 6y = 4
Substitute 2y – 6 for x.
8y – y = 4
Distributive property
14y – 24 = 4
Simplify.
14y = 28
Add 24 to each side.
y = 2
Divide each side by 14.

9. Use the substitution method
EXAMPLE 2
Use the substitution method
STEP 3
Substitute 2 for y in the revised Equation 1 to find the value of x.
x = 2y – 6
Revised Equation 1
x = 2(2) – 6
Substitute 2 for y.
x = –2
Simplify.
ANSWER
The solution is (–2, 2).

10. Use the substitution method
EXAMPLE 2
GUIDED PRACTICE
Use the substitution method
CHECK
Substitute –2 for x and 2 for y in each of the original
equations.
Equation 1
Equation 2
4x + 6y = 4
x – 2y = –6
–2 – 2(2) = –6 ?
4(–2) + 6 (2) = 4 ?
–6 = –6
4 = 4

11. EXAMPLE 1
GUIDED PRACTICE
Use the substitution method
for Examples 1 and 2
Solve the linear system using the substitution method.
y = 2x + 5 1.
3x + y = 10
ANSWER
(1, 7)

12. EXAMPLE 2
GUIDED PRACTICE
Use the substitution method
for Examples 1 and 2
Solve the linear system using the substitution method.
x – y = 3 2.
x + 2y = –6
ANSWER
(0, –3)

13. EXAMPLE 2
GUIDED PRACTICE
Use the substitution method
for Examples 1 and 2
Solve the linear system using the substitution method.
3x + y = –7 3.
–2x + 4y = 0
ANSWER
(–2, –1)

14. EXAMPLE 3
Solve a multi-step problem
WEBSITES
Many businesses pay website hosting companies to store and maintain the computer files that make up their websites. Internet service providers also offer website hosting. The costs for website hosting offered by a website hosting company and an Internet service provider are shown in the table. Find the number of months after which the total cost for website hosting will be the same for both companies.

15. EXAMPLE 3
Solve a multi-step problem
SOLUTION
STEP 1
Write a system of equations. Let y be the total
cost after x months.
Equation 1: Internet service provider
y = x

16. Solve a multi-step problem
EXAMPLE 3
Solve a multi-step problem
Equation 2: Website hosting company
y = x
The system of equations is:
y = x
Equation 1
y = 22.45x
Equation 2

17. Solve a multi-step problem
EXAMPLE 3
Solve a multi-step problem
STEP 2
Substitute 22.45x for y in Equation 1 and solve
for x.
y = x
Write Equation 1.
22.45x = x
Substitute 22.45x for y.
0.5x = 10
Subtract 21.95x from each side.
x = 20
Divide each side by 0.5.
The total cost will be the same for both companies after 20 months.
ANSWER

18. GUIDED PRACTICE
for Example 3 4.
In Example 3, what is the total cost for website hosting for each company after 20 months?
$449
ANSWER

19. GUIDED PRACTICE
for Example 3 5.
WHAT IF? In Example 3, suppose the Internet service provider offers $5 off the set-up fee. After how many months will the total cost for website hosting be the same for both companies?
10 mo
ANSWER

20. EXAMPLE 4
Solve a mixture problem
ANTIFREEZE
For extremely cold temperatures, an automobile manufacturer recommends that a 70% antifreeze and 30% water mix be used in the cooling system of a car. How many quarts of pure (100%) antifreeze and a 50% antifreeze and 50% water mix should be combined to make 11 quarts of a 70% antifreeze and 30% water mix?

21. EXAMPLE 4
Solve a mixture problem
SOLUTION
STEP 1
Write an equation for the total number of quarts and an equation for the number of quarts of antifreeze. Let x be the number of quarts of 100% antifreeze, and let y be the number of quarts of a 50% antifreeze and 50% water mix.

22. EXAMPLE 4
Solve a mixture problem
Equation 1: Total number of quarts
x + y = 11
Equation 2: Number of quarts of antifreeze
x quarts of
100% antifreeze
y quarts of
50%–50% mix
11 quarts of
70%–30% mix
1 x y = (11)
x + 0.5y = 7.7

23. Solve a mixture problem
EXAMPLE 4
Solve a mixture problem
The system of equations is:
x + y =11
Equation 1
x + 0.5y = 7.7
Equation 2
STEP 2
Solve Equation 1 for x.
x + y = 11
Write Equation 1
x = 11 – y
Revised Equation 1
STEP 3
Substitute 11 – y for x in Equation 2 and solve
for y.
x + 0.5y = 7.7
Write Equation 2.

24. Solve a mixture problem
EXAMPLE 4
Solve a mixture problem
(11 – y) + 0.5y = 7.7
Substitute 11 – y for x.
Solve for y.
y = 6.6
STEP 4
Substitute 6.6 for y in the revised Equation 1 to
find the value of x.
x = 11 – y = 11 – 6.6 = 4.4
ANSWER
Mix 4.4 quarts of 100% antifreeze and 6.6 quarts of a 50%
antifreeze and 50% water mix to get 11 quarts of a 70%
antifreeze and 30% water mix.

25. GUIDED PRACTICE
for Example 4
WHAT IF? How many quarts of 100% antifreeze and a 50% antifreeze and 50% water mix should be combined to make 16 quarts of a 70% antifreeze and 30% water mix? 6.
ANSWER
6.4 quarts of 100% antifreeze and 9.6 quarts of a 50%
antifreeze and 50% water mix

26. Warm-up: Homework: Page 381 #2-28 all and #31, 32 and 35

27. Daily Homework Quiz
Solve the linear system using substitution
1. –5x – y = 12
3x – 5y = 4
ANSWER
(–2, –2)
x + 9y = –4
x – 2y = 11
ANSWER
(7, –2)

28. Daily Homework Quiz 3.
You are making 6 quarts of fruit punch for a party. You want the punch to contain 80% fruit juice. You have bottles of 100% fruit juice and 20% fruit juice. How many quarts of 100% fruit juice and how many quarts of 20% fruit juice should you mix to make 6 quarts of 80% fruit juice?
ANSWER
4.5 quarts of 100% fruit juice and 1.5 quarts of 20% fruit juice