Download lesson 7.3

7.3
Solve Linear Systems by Adding or Subtracting
Warm Up
Lesson Presentation
Lesson Quiz
7.3
Warm-Up
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
3h
7.3
Example 1
Solve the linear system:
2x + 3y = 11
–2x + 5y = 13
Equation 1
Equation 2
SOLUTION
STEP 1
STEP 2
Add the equations to
eliminate one variable.
Solve for y.
2x + 3y = 11
–2x + 5y = 13
8y = 24
y=3
7.3
Example 1
STEP 3
Substitute 3 for y in either equation and
solve for x.
2x + 3y = 11
2x + 3(3) = 11
x=1
ANSWER
The solution is (1, 3).
Write Equation 1
Substitute 3 for y.
Solve for x.
7.3
Example 1
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
7.3
Example 2
Solve the linear system:
Equation 1
4x + 3y = 2
5x + 3y = –2
Equation 2
SOLUTION
STEP 1
Subtract the equations to
eliminate one variable.
STEP 2
Solve for x.
4x + 3y = 2
5x + 3y = –2
–x
= 4
x = 4
7.3
Example 2
STEP 3 Substitute 4 for x in either equation and solve
for y.
4x + 3y = 2
4(–4) + 3y = 2
y=6
Write Equation 1.
Substitute –4 for x.
Solve for y.
ANSWER
The solution is (–4, 6).
7.3
Example 3
Solve the linear system: 8x – 4y = –4
4y = 3x + 14
Equation 1
Equation 2
SOLUTION
STEP 1
STEP 2
STEP 3
Rewrite Equation 2 so that the like terms are
arranged in columns.
8x – 4y = –4
8x – 4y = –4
4y = 3x + 14
3x + 4y = 14
5x
= 10
Add the equations.
Solve for x.
x=2
7.3
Example 3
STEP 4
Substitute 2 for x in either equation and
solve for y.
4y = 3x + 14
4y = 3(2) + 14
y=5
Write Equation 2.
Substitute 2 for x.
Solve for y.
ANSWER
The solution is (2, 5).
7.3
Guided Practice
Solve the linear system.
1.
4x – 3y = 5
–2x + 3y = –7
ANSWER
3.
(–1, –3)
6x – 4y = 14
– 3x + 4y = 1
ANSWER
5.
2.
(5, 4)
3x + 4y = –6
2y = 3x + 6
ANSWER
(–2, 0)
– 5x – 6y = 8
5x + 2y = 4
ANSWER
4.
7x – 2y = 5
7x – 3y = 4
ANSWER
6.
(2, –3)
(1, 1)
2x + 5y = 12
5y = 4x + 6
ANSWER
(1, 2)
7.3
Example 4
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.
7.3
Example 4
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
7.3
Example 4
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
Equation 2: Going downstream
x
+
y
=
6
7.3
Example 4
STEP 2
Solve the system of equations.
x–y=4
Write Equation 1.
x+y=6
Write Equation 2.
2x
= 10
x=5
Add equations.
Solve for x.
Substitute 5 for x in Equation 2 and solve for y.
Substitute 5 for x in Equation 2.
5+y=6
y=1
Subtract 5 from each side.
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.
7.3
Guided Practice
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.
7.
ANSWER
average speed of the kayak: 3.5 mi/h, speed of the
current 1.5 mi/h
7.3
Lesson Quiz
Solve the linear system using elimination.
1.
–5x + y = 18
3x – y = –10
ANSWER
3.
2. 4x + 2y = 14
4x – 3y = –11
(–4, –2)
2x – y = –14
y = 3x + 6
ANSWER
ANSWER
4.
(8, 30)
(1, 5)
x + 4y = 15
2y = x – 9
ANSWER
(11, 1)
7.3
Lesson Quiz
5. 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.
ANSWER
flat fee: $2, price per page: $0.84