Transcript 2 - SVHSAlgebra1

Warm-Up Exercises
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
EXAMPLE
Warm-Up1Exercises
Use addition to eliminate a variable
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
EXAMPLE
Warm-Up1Exercises
Use addition to eliminate a variable
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.
EXAMPLE
Warm-Up1Exercises
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
EXAMPLE
Warm-Up2Exercises
Use subtraction to eliminate a variable
Solve the linear system:
4x + 3y = 2
5x + 3y = –2
Equation 1
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
EXAMPLE
Warm-Up2Exercises
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.
Substitute –4 for x.
4(–4) + 3y = 2
y=6
Solve for y.
ANSWER
The solution is (–4, 6).
EXAMPLE
Warm-Up3Exercises
Arrange like terms
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
EXAMPLE
Warm-Up3Exercises
Arrange like terms
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).
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1,2 and 3
Solve the linear system:
1.
4x – 3y = 5
–2x + 3y = –7
ANSWER
(–1, –3)
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1,2 and 3
Solve the linear system:
2. – 5x – 6y = 8
5x + 2y = 4
ANSWER
(2, –3)
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1,2 and 3
Solve the linear system:
3.
6x – 4y = 14
– 3x + 4y = 1
ANSWER
(5, 4)
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1,2 and 3
Solve the linear system:
4.
7x – 2y = 5
7x – 3y = 4
ANSWER
(1, 1)
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1,2 and 3
Solve the linear system:
5.
3x + 4y = –6
2y = 3x + 6
ANSWER
(–2, 0)
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1,2 and 3
Solve the linear system:
6.
2x + 5y = 12
5y = 4x + 6
ANSWER
(1, 2)
Warm-Up4Exercises
EXAMPLE
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.
Warm-Up4Exercises
EXAMPLE
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
Warm-Up4Exercises
EXAMPLE
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
Warm-Up4Exercises
EXAMPLE
Write and solve a linear system
Equation 2: Going downstream
x
+
y
=
6
Warm-Up4Exercises
EXAMPLE
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
x=5
Add equations.
Solve for x.
Substitute 5 for x in Equation 2 and solve for y.
Warm-Up4Exercises
EXAMPLE
Write and solve a linear system
5+y=6
y=1
Substitute 5 for x in Equation 2.
Subtract 5 from each side.
Warm-Up
Exercises
GUIDED
PRACTICE
7.
for Example 4
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
Daily
Homework
Quiz
Warm-Up
Exercises
Solve the linear system using elimination.
1.
–5x + y = 18
3x – y = –10
ANSWER
2.
4x + 2y = 14
4x – 3y = –11
ANSWER
3.
(–4, –2)
(1, 5)
2x – y = –14
y = 3x + 6
ANSWER
(8, 30)
Daily
Homework
Quiz
Warm-Up
Exercises
4.
x + 4y = 15
2y = x – 9
ANSWER
(11, 1)
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: $.84