Transcript 2-8 Solving Two-Step Equations

2-8
2-8 Solving
SolvingTwo-Step
Two-StepEquations
Equations
Warm Up
Problem of the Day
Lesson Presentation
Course
Course
33
2-8 Solving Two-Step Equations
Warm Up
Solve.
1. x + 12 = 35 x = 23
2. 8x = 120
x = 15
3. y = 7
y = 63
9
4. –34 = y + 56 y = –90
Course 3
2-8 Solving Two-Step Equations
Problem of the Day
x is an odd integer. If you triple x and
then subtract 7, you get a prime
number. What is the smallest possible
x? (Hint: What is the smallest prime
number?)
x=3
Course 3
2-8 Solving Two-Step Equations
Learn to solve two-step equations.
Course 3
2-8 Solving Two-Step Equations
Sometimes more than one inverse operation
is needed to solve an equation. Before
solving, ask yourself, “What is being done to
the variable, and in what order?” Then work
backward to undo the operations.
Course 3
2-8 Solving Two-Step Equations
Additional Example 1: Problem Solving Application
The mechanic’s bill to repair Mr. Wong’s
car was $650. The mechanic charges
$45 an hour for labor, and the parts that
were used cost $443. How many hours
did the mechanic work on the car?
Course 3
2-8 Solving Two-Step Equations
Additional Example 1 Continued
1
Understand the Problem
List the important information:
The answer is the number of hours the
mechanic worked on the car.
• The parts cost $443.
• The labor cost $45 per hour.
• The total bill was $650.
Let h represent the hours the mechanic
worked.
Total bill =
Parts
+
Labor
650
=
443
+
45h
Course 3
2-8 Solving Two-Step Equations
Additional Example 1 Continued
2
Make a Plan
Think: First the variable is multiplied by
45, and then 443 is added to the result.
Work backward to solve the equation.
Undo the operations in reverse order:
First subtract 443 from both sides of the
equation, and then divide both sides of
the new equation by 45.
Course 3
2-8 Solving Two-Step Equations
Additional Example 1 Continued
3
Solve
650 = 443 + 45h
–443 –443
207 =
207 = 45h
45
45
Subtract to undo the addition.
45h
Divide to undo multiplication.
4.6 = h
The mechanic worked for 4.6 hours on Mr. Wong’s
car.
Course 3
2-8 Solving Two-Step Equations
Additional Example 1 Continued
4
Look Back
You can use a table to decide whether your
answer is reasonable.
Hours Labor
1
45
Parts
$443
Total Cost
$488
2
3
4
90
135
180
$443
$443
$443
$533
$578
$623
5
225
$443
$668
4.6 hours is a reasonable answer.
Course 3
2-8 Solving Two-Step Equations
Check It Out: Example 1
The mechanic’s bill to repair your car
was $850. The mechanic charges $35
an hour for labor, and the parts that
were used cost $275. How many
hours did the mechanic work on your
car?
Course 3
2-8 Solving Two-Step Equations
Check It Out: Example 1 Continued
1
Understand the Problem
List the important information:
The answer is the number of hours the
mechanic worked on your car.
• The parts cost $275.
• The labor cost $35 per hour.
• The total bill was $850.
Let h represent the hours the mechanic worked.
Total bill =
Parts
+
Labor
850
=
275
+
35h
Course 3
2-8 Solving Two-Step Equations
Check It Out: Example 1 Continued
2
Make a Plan
Think: First the variable is multiplied by 35,
and then 275 is added to the result. Work
backward to solve the equation. Undo the
operations in reverse order: First subtract
275 from both sides of the equation, and
then divide both sides of the new equation by
35.
Course 3
2-8 Solving Two-Step Equations
Check It Out: Example 1 Continued
3
Solve
850 = 275 + 35h
–275 –275
575 =
575 = 35h
35
35
Subtract to undo the addition.
35h
Divide to undo multiplication.
16.4  h
The mechanic worked for about 16.4 hours on your
car.
Course 3
2-8 Solving Two-Step Equations
Check It Out: Example 1 Continued
4
Look Back
You can use a table to decide whether your
answer is reasonable.
Hours Labor
13
455
Parts
$275
Total Cost
$730
14
15
16
490
525
560
$275
$275
$275
$765
$800
$835
17
595
$275
$870
16.4 hours is a reasonable answer.
Course 3
2-8 Solving Two-Step Equations
Additional Example 2A: Solving Two-Step Equations
n
Solve
+ 7 = 22
3
Method 1: Work backward to isolate the variable.
Think: First the variable is divided by 3, and then 7
is added. To isolate the variable, subtract 7, and
then multiply by 3.
n
+ 7 – 7 = 22 – 7
3
n
3
=3
3


n = 45
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15
Subtract 7 from both sides.
Multiply both sides by 3.
2-8 Solving Two-Step Equations
Additional Example 2A Continued
Solve n + 7 = 22
3
Method 2: Multiply both sides of the equation by the
denominator.
(3) n + 7 = 22(3)
3
Multiply both sides by the
denominator.
n + 21 = 66
–21
n
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–21
= 45
Subtract to undo addition.
2-8 Solving Two-Step Equations
Additional Example 2B: Solving Two-Step Equations
Solve y – 4 = 9
3
Method 1: Work backward to isolate the variable.
y
4
Rewrite the expression as
–
=9
3
3
the sum of two fractions.
4
Think: First the variable is divided by 3, and then is
3
4
subtracted. To isolate the variable, add and then
3
multiply by 3.
4
y
4
4
4
Add to both sides.
–
+
=9+
3
3
3
3
3
31
y
(3)
=
(3)
Multiply both sides by 3.
t3
3
y = 31
Course 3
2-8 Solving Two-Step Equations
Additional Example 2B: Solving Two-Step Equations
Solve y – 4 = 9
3
Method 2: Multiply both sides of the equation by
the denominator.
y – 4= 9
3
(3) y – 4 = 9(3)
3
Multiply both sides by the
denominator.
y – 4 = 27
+4
+4
y = 31
Course 3
Add to undo subtraction.
2-8 Solving Two-Step Equations
Check It Out: Example 2A
n
+ 8 = 18
4
Method 1: Work backward to isolate the variable.
Think: First the variable is divided by 4, and then 8
is added. To isolate the variable, subtract 8, and
then multiply by 4.
Solve
n
+ 8 – 8 = 18 – 8
4
n
4
=4
4


n = 40
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10
Subtract 8 from both sides.
Multiply both sides by 4.
2-8 Solving Two-Step Equations
Check It Out: Example 2A
Solve
n
+ 8 = 18
4
Method 2: Multiply both sides of the equation by the
denominator.
(4) n + 8 = 18(4)
4
Multiply both sides by the
denominator.
n + 32 = 72
–32
n
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–32
= 40
Subtract to undo addition.
2-8 Solving Two-Step Equations
Check It Out: Example 2B
Solve y – 7 = 7
2
Method 1: Work backward to isolate the variable.
y
7
Rewrite the expression as
–
=7
2
2
the sum of two fractions.
7
Think: First the variable is divided by 2, and then is
2
7
subtracted. To isolate the variable, add and then
2
multiply by 2.
7
y
7
7
7
Add to both sides.
–
+
=7+
2
2
2
2
2
21
y
(2)
=
(2)
Multiply both sides by 2.
t2
2
y = 21
Course 3
2-8 Solving Two-Step Equations
Check It Out: Example 2B
Solve y – 7 = 7
2
Method 2: Multiply both sides of the equation by
the denominator.
y – 7= 7
2
(2) y – 7 = 7(2)
2
Multiply both sides by the
denominator.
y – 7 = 14
+7
+7
y = 21
Course 3
Add to undo subtraction.
2-8 Solving Two-Step Equations
Lesson Quiz
Solve.
1. x – 3 = 10
–9
2. 7y + 25 = –24
x = –117
y = –7
3. –8.3 = –3.5x + 13.4 x = 6.2
4. y + 5 = 3
11
y = 28
5. The cost for a new cell phone plan is $39 per
month plus a one-time start-up fee of $78. If
you are charged $1014, how many months will
the contract last? 24
Course 3