1. 1. Solve the linear system using substitution.
2x + y = 12
3x – 2y = 11
ANSWER
(5, 2)
2. One auto repair shop charges $30 for a diagnosis and $25 per hour for labor. Another auto repair shop charges $35 per hour for labor. For how many hours are the total charges for both of the shops the same?
ANSWER
3 h

2. Use addition to eliminate a variable
EXAMPLE 1
Use addition to eliminate a variable
Solve the linear system:
2x + 3y = 11
Equation 1
–2x + 5y = 13
Equation 2
SOLUTION
STEP 1
Add the equations to
eliminate one variable.
2x + 3y = 11
–2x + 5y = 13
STEP 2
Solve for y.
8y = 24
y = 3

3. Use addition to eliminate a variable
EXAMPLE 1
Use addition to eliminate a variable
STEP 3
Substitute 3 for y in either equation and
solve for x.
2x + 3y = 11
Write Equation 1
2x + 3(3) = 11
Substitute 3 for y.
x = 1
Solve for x.
ANSWER
The solution is (1, 3).

4. Use addition to eliminate a variable
EXAMPLE 1
Use addition to eliminate a variable
CHECK
Substitute 1 for x and 3 for y in each of the original equations.
2x + 3y = 11
−2x + 5y = 13
2(1) + 3(3) = 11 ?
−2(1) + 5(3) = 13 ?
11 = 11
13 = 13

5. Use subtraction to eliminate a variable
EXAMPLE 2
Use subtraction to eliminate a variable
Solve the linear system:
4x + 3y = 2
Equation 1
5x + 3y = –2
Equation 2
SOLUTION
STEP 1
Subtract the equations to
eliminate one variable.
4x + 3y = 2
5x + 3y = –2
STEP 2
Solve for x.
– x = 4
x = −4

6. Use subtraction to eliminate a variable
EXAMPLE 2
Use subtraction to eliminate a variable
STEP 3
Substitute −4 for x in either equation and solve
for y.
4x + 3y = 2
Write Equation 1.
4(–4) + 3y = 2
Substitute –4 for x.
y = 6
Solve for y.
ANSWER
The solution is (–4, 6).

7. Solve the linear system: 8x – 4y = –4 4y = 3x + 14
EXAMPLE 3
Arrange like terms
Solve the linear system:
8x – 4y = –4
Equation 1
4y = 3x + 14
Equation 2
SOLUTION
STEP 1
Rewrite Equation 2 so that the like terms are
arranged in columns.
8x – 4y = –4
8x – 4y = –4
4y = 3x + 14
−3x + 4y = 14
STEP 2
Add the equations.
5x = 10
STEP 3
Solve for x.
x = 2

8. Substitute 2 for x in either equation and solve for y.
EXAMPLE 3
Arrange like terms
STEP 4
Substitute 2 for x in either equation and
solve for y.
4y = 3x + 14
Write Equation 2.
4y = 3(2) + 14
Substitute 2 for x.
y = 5
Solve for y.
ANSWER
The solution is (2, 5).

9. GUIDED PRACTICE
for Example 1,2 and 3
Solve the linear system: 1.
4x – 3y = 5 `
–2x + 3y = –7
ANSWER
(–1, –3)

10. GUIDED PRACTICE
for Example 1,2 and 3
Solve the linear system: 2.
5x – 6y = 8 –
5x + 2y = 4
ANSWER
(2, –3)

11. GUIDED PRACTICE
for Example 1,2 and 3
Solve the linear system: 3.
6x – 4y = 14
3x + 4y = 1 –
ANSWER
(5, 4)

12. GUIDED PRACTICE
for Example 1,2 and 3
Solve the linear system: 4.
7x – 2y = 5
7x – 3y = 4
ANSWER
(1, 1)

13. GUIDED PRACTICE
for Example 1,2 and 3
Solve the linear system: 5.
3x + 4y = –6 =
3x + 6 2y
ANSWER
(–2, 0)

14. GUIDED PRACTICE
for Example 1,2 and 3
Solve the linear system: 6.
2x + 5y = 12 =
4x + 6 5y
ANSWER
(1, 2)

15. EXAMPLE 4
Write and solve a linear system
KAYAKING
During a kayaking trip, a kayaker travels 12 miles upstream (against the current) and 12 miles downstream (with the current), as shown. The speed of the current remained constant during the trip. Find the average speed of the kayak in still water and the speed of the current.

16. EXAMPLE 4
Write and solve a linear system
STEP 1
Write a system of equations. First find the speed of the kayak going upstream and the speed of the kayak going downstream.
Upstream:
d = rt
Downstream:
d = rt
12 = r 3
12 = r 2
4 = r
6 = r

17. EXAMPLE 4
Write and solve a linear system
Use the speeds to write a linear system. Let x be the average speed of the kayak in still water, and let y be the speed of the current.
Equation 1:
Going upstream x y 4 = –

18. EXAMPLE 4
Write and solve a linear system
Equation 2:
Going downstream x y 6 = +

19. Write and solve a linear system
EXAMPLE 4
Write and solve a linear system
STEP 2
Solve the system of equations.
x – y = 4
Write Equation 1.
x + y = 6
Write Equation 2.
2x = 10
Add equations.
x = 5
Solve for x.
Substitute 5 for x in Equation 2 and solve for y.

20. Write and solve a linear system
EXAMPLE 4
Write and solve a linear system
5 + y = 6
Substitute 5 for x in Equation 2.
y = 1
Subtract 5 from each side.
ANSWER
The average speed of the kayak in still water is 5 miles per hour, and the speed of the current is 1 mile per hour.

21. GUIDED PRACTICE
for Example 4
7. WHAT IF? In Example 4, suppose it takes the kayaker 5 hours to travel 10 miles upstream and 2 hours to travel 10 miles downstream. The speed of the current remains constant during the trip. Find the average speed of the kayak in still water and the speed of the current.
ANSWER
average speed of the kayak: 3.5 mi/h, speed of the current: 1.5 mi/h

22. Daily Homework Quiz
Solve the linear system using elimination.
1. –5x + y = 18
3x – y = –10
ANSWER
(–4, –2)
x + 2y = 14
4x – 3y = –11
ANSWER
(1, 5)
x – y = –14
y = 3x + 6
ANSWER
(8, 30)

23. Daily Homework Quiz
4. x + 4y = 15
2y = x – 9
ANSWER
(11, 1)
A business center charges a flat fee to send faxes plus a fee per page. You send one fax with 4 pages for $5.36 and another fax with 7 pages for $7.88. Find the flat fee and the cost per page to send a fax. 5.
ANSWER
flat fee: $2, price per page: $.84