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Warm-Up Exercises
1. Graph the equation –2x + y = 1.
ANSWER
2. It takes 3 hours to mow a lawn and 2 hours to trim
hedges. You spend 16 hours doing yard work.
What are 2 possible numbers of lawns you mowed
and hedges you trimmed?
ANSWER
2 lawns and 5 hedges, or
4 lawns and 2 hedges
Check the intersection point
Warm-Up1Exercises
EXAMPLE
Use the graph to solve the
system. Then check your
solution algebraically.
x + 2y = 7
Equation 1
3x – 2y = 5
Equation 2
SOLUTION
The lines appear to intersect at the point (3, 2).
CHECK Substitute 3 for x and 2 for y in each equation.
x + 2y = 7
?
3 + 2(2) = 7
7=7
Check the intersection point
Warm-Up1Exercises
EXAMPLE
3x – 2y = 5
?
3(3) – 2(2) = 5
5=5
ANSWER
Because the ordered pair (3, 2) is a solution of each
equation, it is a solution of the system.
Use the graph-and-check method
Warm-Up2Exercises
EXAMPLE
Solve the linear system:
–x + y = –7
Equation 1
x + 4y = –8
Equation 2
SOLUTION
STEP 1
Graph both equations.
Use the graph-and-check method
Warm-Up2Exercises
EXAMPLE
STEP 2
Estimate the point of intersection. The two lines
appear to intersect at (4, – 3).
STEP 3
Check whether (4, –3) is a solution by substituting 4 for
x and –3 for y in each of the original equations.
Equation 1
–x + y = –7
?
–(4) + (–3) = –7
–7 = –7
Equation 2
x + 4y = –8
?
4 + 4(–3) = –8
–8 = –8
Use the graph-and-check method
Warm-Up2Exercises
EXAMPLE
ANSWER
Because (4, –3) is a solution of each equation, it is a
solution of the linear system.
Use the graph-and-check
method
Warm-Up
EXAMPLE
2Exercises
for Examples 1 and
2
GUIDED
PRACTICE
Solve the linear system by graphing. Check your solution.
1. –5x + y = 0
5x + y = 10
ANSWER
(1, 5)
Use the graph-and-check
method
Warm-Up
EXAMPLE
2Exercises
for Examples 1 and
2
GUIDED
PRACTICE
Solve the linear system by graphing. Check your solution.
2. –x + 2y = 3
2x + y = 4
ANSWER
(1, 2)
Use the graph-and-check
method
Warm-Up
EXAMPLE
2Exercises
for Examples 1 and
2
GUIDED
PRACTICE
Solve the linear system by graphing. Check your solution.
3. x – y = 5
3x + y = 3
ANSWER
(2, –3)
Warm-Up3Exercises
EXAMPLE
Standardized Test Practice
The parks and recreation department in your town
offers a season pass for $90.
•
As a season pass holder, you pay $4 per session
to use the town’s tennis courts.
•
Without the season pass, you pay $13 per session
to use the tennis courts.
Warm-Up3Exercises
EXAMPLE
Standardized Test Practice
Which system of equations can be used to find the
number x of sessions of tennis after which the total
cost y with a season pass, including the cost of the
pass, is the same as the total cost without a season
pass?
A y = 4x
y = 13x
B y = 4x
y = 90 + 13x
C y = 13x
y = 90 + 4x
D y = 90 + 4x
y = 90 + 13x
Warm-Up3Exercises
EXAMPLE
Standardized Test Practice
SOLUTION
Write a system of equations where y is the total cost
(in dollars) for x sessions.
EQUATION 1
y
=
13
x
Warm-Up3Exercises
EXAMPLE
Standardized Test Practice
EQUATION 2
y
=
90
+
4
x
ANSWER
The correct answer is C.
A
B
C
D
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 3
4. Solve the linear system in Example 3 to find the
number of sessions after which the total cost with a
season pass, including the cost of the pass, is the
same as the total cost without a season pass.
ANSWER
10 sessions
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 3
5. WHAT IF? In Example 3, suppose a season pass
costs $135. After how many sessions is the total cost
with a season pass, including the cost of the pass, the
same as the total cost without a season pass?
ANSWER
15 sessions
Solve a multi-step problem
Warm-Up4Exercises
EXAMPLE
RENTAL BUSINESS
A business rents in-line skates
and bicycles. During one day,
the business has a total of 25
rentals and collects $450 for the
rentals. Find the number of pairs
of skates rented and the number
of bicycles rented.
Solve a multi-step problem
Warm-Up4Exercises
EXAMPLE
SOLUTION
STEP 1
Write a linear system. Let x be the number of pairs
of skates rented, and let y be the number of
bicycles rented.
x + y = 25
Equation for number of rentals
15x + 30y = 450 Equation for money collected from rentals
STEP 2
Graph both equations.
Solve a multi-step problem
Warm-Up4Exercises
EXAMPLE
STEP 3
Estimate the point of
intersection. The two lines
appear to intersect at (20, 5).
STEP 4
Check whether (20, 5) is a
solution.
20 + 5 =? 25
25 = 25
15(20) + 30(5) =? 450
450 = 450
ANSWER
The business rented 20 pairs of skates and 5 bicycles.
Solve a for
multi-step
problem
Warm-Up
EXAMPLE
4Exercises
Example
4
GUIDED
PRACTICE
6. WHAT IF? In Example 4, suppose the business has
a total of 20 rentals and collects $420. Find the
number of bicycles rented.
ANSWER
8 bicycles
Warm-Up Exercises
Warm-up:
Homework: Page 372 # 3-5 all, #12-26 all, # 31,
#33, #35
Daily
Homework
Quiz
Warm-Up
Exercises
1.
Use the graph to solve the linear system
x–y=5
3x + y = 3
ANSWER
(2, –3)
Daily
Homework
Quiz
Warm-Up
Exercises
2.
Solve the linear system by graphing.
2x + y = –3
–6x + 3y = 3
ANSWER
(–1, –1)
Daily
Homework
Quiz
Warm-Up
Exercises
3.
A pet store sells angel fish for $6 each and clown
loaches for $4 each. If the pet store sold 8 fish for
$36, how many of each type of fish did it sell?
ANSWER
2 angel fish and 6 clown loaches