Check - Waynesville R

Transcription

2-3 Solving Multi-Step Equations
Solve each equation. Check your solution.
3m + 4 = 11
Check:
12 = 7f
9
Check:
eSolutions Manual - Powered by Cognero
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NUMBER THEORY Twelve decreased by twice a number equals 34. Write an equation for this situation and
then find the number.
Let n = a number.
Twelve decreased
twice a
by
number
12
2n
equals
34.
=
34
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NUMBER THEORY Twelve decreased by twice a number equals 34. Write an equation for this situation and
then find the number.
Let n = a number.
Twelve decreased
by
12
The equation is 12
twice a
number
2n
equals
34.
=
34
2n = 34, and the number is 23.
BASEBALL Among the career home run leaders for Major League Baseball, Hank Aaron has 175 fewer than
twice the number that Dave Winfield has. Hank Aaron hit 755 home runs. Write an equation for this situation. How
many home runs did Dave Winfield hit in his career?
Let h = the number of home runs Dave Winfield hit.
175 fewer than
twice the
number that
Dave Winfield
has
2h 175
equals
the number of
home runs
Hank Aaron
has
=
755
Dave Winfield hit 465 home runs in his career.
Write an equation and solve each problem.
Find three consecutive odd integers with a sum of 75.
Let n = the least odd integer. Then n + 2 = the next greater odd integer, and n + 4 = the greatest of the three
integers. The sum of the three consecutive odd integers is 75. So, n + (n + 2) + (n + 4) = 75.
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Dave Winfield hit 465 home runs in his career.
Find three consecutive odd integers with a sum of 75.
The integers are 23, 25, and 27.
Find three consecutive integers with a sum of 36.
Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The
sum of the three consecutive integers is 36. So, n + (n + 1) + (n + 2) = 36.
3t + 7 = 8
Check:
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3t + 7 = 8
Check:
8 = 16 + 8n
Check:
34 = 6m
Check:
4
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34 = 6m
4
Check:
9x + 27 = 72
Check:
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FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the
business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many
minutes per month he can use the phone and stay within budget.
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FINANCIAL LITERACY The Cell+ Cellular Phone store offers the plans shown in the table. Raul chose the
business plan and has budgeted $100 per month. Write an equation for this situation, and determine how many
minutes per month he can use the phone and stay within budget.
Let m = the number of minutes Raul uses the phone in a month. The monthly fee for the business plan is $49.99 and
the cost per minute is $0.15. So, 0.15m + 49.99 = 100.
Raul could use the phone an additional 333 minutes per month and stay within budget. The plan gives him 650 free
minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.
Fourteen less than three fourths of a number is negative eight. Find the number.
Let n = the number.
Fourteen
is
less than
three
fourths of a
number
=
14
negative
eight.
8
The number is 8.
Seventeen is thirteen subtracted from six times a number. What is the number?
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2-3 Solving
Multi-Step
Equations
Raul could
use the phone
an additional 333 minutes per month and stay within budget. The plan gives him 650 free
minutes, so the total number of minutes is 650 + 333.4 or about 983 minutes.
Fourteen less than three fourths of a number is negative eight. Find the number.
Let n = the number.
Fourteen
is
less than
three
fourths of a
number
=
14
negative
eight.
8
The number is 8.
Seventeen is thirteen subtracted from six times a number. What is the number?
Let x = the number.
Seventeen
is
17
=
thirteen
subtracted
from six
times a
number.
6x 13
The number is 5.
Find three consecutive even integers with the sum of 84.
Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three
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The number is 5.
Find three consecutive even integers with the sum of 84.
Let n = the least even integer. Then n + 2 = the next greater even integer, and n + 4 = the greatest of the three
integers. The sum of the three consecutive even integers is 84. So, n + (n + 2) + (n + 4) = 84.
Find three consecutive odd integers with the sum of 141.
Find four consecutive integers with the sum of 54.
Let n = the least integer. Then n + 1 = the next greater integer, n + 2 = the next integer, and n + 3 = the greatest of
the integers. The sum of the three consecutive integers is 54. So, n + (n + 1) + (n + 2) + (n + 3) = 54.
The integers are 12, 13, 14, and 15.
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the integers. The sum of the three consecutive integers is 142. So, + (n + 1) + (n + 2) + (n + 3) = 142.
6m 8 = 24
Check:
45 = 7
5n
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45 = 7
5n
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CCSS REASONING The ages of three brothers are consecutive integers with the sum of 96. How old are the
brothers?
Let n = the least integer. Then n + 1 = the next greater integer, and n + 2 = the greatest of the three integers. The
sum of the three consecutive integers is 96. So, n + n + 1 + n + 2 = 96.
The brothers are 31, 32, and 33.
VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a
seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lavaPage 23
flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?
2-3 Solving
Multi-Step
Equations
The brothers
are 31, 32,
and 33.
VOLCANOES Moving lava can build up and form beaches at the coast of an island. The growth of an island in a
seaward direction may be modeled as 8y + 2 centimeters, where y represents the number of years that the lava
flows. An island has expanded 60 centimeters seaward. How long has the lava flowed?
To find how long the lava has flowed if the island has expanded 60 centimeters, solve 8y + 2 = 60 for y.
The lava has flowed
years or 7 years and 3 months.
5x 4.8 = 6.7
Check:
3.7q + 26.2 = 111.67
Check:
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3.7q + 26.2 = 111.67
Check:
0.6a + 9 = 14.4
Check:
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Check:
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3.6
2.4m = 12
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3.6
2.4m = 12
Check:
If 7m
3 = 53, what is the value of 11m + 2?
To find the value of 11m + 2, first solve 7m
3 = 53 to find the value of m.
Now, replace m with 8 in the expression 11m + 2.
So, 11m + 2 = 90.
If 13y + 25 = 64, what is the value of 4y
To find the value of 4y
7?
7, first solve 13y + 25 = 64 for y.
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Now, replace y with 3 in the expression 4y
7.
So, 11m + 2 = 90.
If 13y + 25 = 64, what is the value of 4y
To find the value of 4y
7?
7, first solve 13y + 25 = 64 for y.
Now, replace y with 3 in the expression 4y
So, 4y
7.
7 = 5.
If 5c + 6 = 69, what is the value of 6c
To find the value of 6c
15?
15, first solve 5c + 6 = 69 for c.
Now, replace c with 15 in the expression 6c
So, 6c
15.
15 = 75.
AMUSEMENT PARKS An amusement park offers a yearly membership of $275 that allows for free parking and
admission to the park. Members can also use the water park for an additional $5 per day. Nonmembers pay $6 for
parking, $15 for admission, and $9 for the water park.
a. Write and solve an equation to find the number of visits it would take for the total cost to be the same for a
member and a nonmember if they both use the water park at each visit.
b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park.
c. Plot these points on a coordinate graph and describe what you see.
a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The
is represented by the expression x(6 + 15 + 9). To find the number of visits it Page 28
would take for the total cost to be the same for a member and a nonmember if they both use the water park at each
visit, set the two expressions equal to each other and solve for x.
eSolutions
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cost Manual
for x visits
for by
a nonmember
member and a nonmember if they both use the water park at each visit.
b. Make a table for the costs of members and nonmembers after 3, 6, 9, 12, and 15 visits to the park.
c. Plot these points on a coordinate graph and describe what you see.
a. Let x = the number of visits. The cost for x visits for a member is represented by the expression 5x + 275. The
cost for x visits for a nonmember is represented by the expression x(6 + 15 + 9). To find the number of visits it
would take for the total cost to be the same for a member and a nonmember if they both use the water park at each
visit, set the two expressions equal to each other and solve for x.
The total cost would be the same for a member and a nonmember if they both use the water park at each visit for 11
visits.
b.
Visits
3
6
9
12
15
Cost for
Members
5(3) + 275 =
290
5(6) + 275 =
305
5(9) + 275 =
320
5(12) + 275 =
335
5(15) + 275 =
350
Cost for
Nonmembers
3(6 + 15 + 9) = 90
6(6 + 15 + 9) = 180
9(6 + 15 + 9) = 270
12(6 + 15 + 9) =
360
15(6 + 15 + 9) =
450
c. Graph the number of visits on the x-axis and the cost on the y-axis. Then graph the ordered pairs from the table.
Use a different colored point for the members and nonmembers.
Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11.
Therefore, if a person is going to visit the park less than 11 times, it will be cheaper to be a nonmember.
SHOPPING At The Family Farm, you can pick your own fruits and vegetables.
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Both functions are linear. The points for nonmembers are lower than the points for members when x is less than 11.
Therefore,
if a personEquations
is going to visit the park less than 11 times, it will be cheaper to be a nonmember.
2-3 Solving
Multi-Step
SHOPPING At The Family Farm, you can pick your own fruits and vegetables.
a. The cost of a bag of potatoes is $1.50 less than
of the price of apples. Write and solve an equation to find the
cost of potatoes.
b. The price of each zucchini is 3 times the price of winter squash minus $7. Write and solve an equation to find the
cost of zucchini.
c. Write an equation to represent the cost of a pumpkin using the cost of the blueberries.
a. Let a = the cost of a bag of apples and p
cost of a bag of potatoes.
The cost
of a bag
of
potatoes
is
p
=
$1.50 less than
of the price
of apples.
The cost of a bag of potatoes is about $2.00.
b. Let z = the price of zucchini and w = the price of winter squash.
The price
is
3 times
minus
$7.
of each
the
zucchini
price of
winter
squash
z
=
3w
7
The Manual
cost of- zucchini
$1.97.
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c. Let p = the cost of a pumpkin and b = the cost of blueberries.
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The cost of zucchini is $1.97.
c. Let p = the cost of a pumpkin and b = the cost of blueberries.
The cost
of a
pumpkin
is
2
p
=
2
times
the
minus
cost
of
blueberries
b
0.98
0.98
An equation that represents the cost of a pumpkin using the cost of the blueberries is p = 2b
0.98.
OPEN ENDED Write a problem that can be modeled by the equation 2x + 40 = 60. Then solve the equation and
explain the solution in the context of the problem.
Sample answer: A pair of designer jeans costs $60. This is $40 more than twice the cost of a T shirt. How much is
the T shirt?
The T shirt costs $10.
CHALLENGE Solve each equation for x. Assume that a
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The T shirt costs $10.
CHALLENGE Solve each equation for x. Assume that a
Determine whether each equation has a solution. Justify your answer.
a.
b.
c.
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a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we
subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have
Determine whether each equation has a solution. Justify your answer.
a.
b.
c.
a. For any fraction to equal 1, the numerator and denominator must be equal. So, a + 4 must equal a + 5. If we
subtract a from each side, we are left with 4 = 5 which is impossible. Therefore, the original equation does not have
a solution.
b. For any fraction to equal 1, the numerator and denominator must be equal. So, 1 + b must equal 1 b. If we
subtract 1 from each side, we are left with b = b which is true only when b = 0. Therefore, the equation has a
solution, 0.
c. For any fraction to equal 1, the numerator and denominator must be equal. So, c 5 must equal 5 c. If we add c
+ 5 to each side, we are left with 2c = 10 which reduces to c = 5. However, when c equals 5, the original fraction
becomes
or which is undefined. Therefore, the original equation does not have a solution.
CCSS REGULARITY Determine whether the following statement is sometimes, always, or never true. Explain
your reasoning.
The sum of three consecutive odd integers equals an even integer.
The statement is never true. Whenever three odd integers are added together, the sum is always odd. The first two
odd numbers will always sum to an even number, and the sum of this even number and the third odd number will
Test a few examples:
3 + 5 + 7 = 15
9 + 13 + 17 = 39
11 + 19 + 33 = 63
The algebraic proof of this statement is beyond the scope of this course.
WRITING IN MATH Write a paragraph explaining the order of the steps that you would take to solve a multi-step
equation.
Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example,
in order to solve the equation 4k + 20 = 236, you would first subtract 20 from each side and then divide each side by
4.
Which is the best estimate for the number of minutes on the calling card advertised below?
A 10 min
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Sample answer: To solve a linear equation, first isolate the variable term. Then, solve for the variable. For example,
in order Multi-Step
to solve the equation
4k + 20 = 236, you would first subtract 20 from each side and then divide each side by
2-3 Solving
Equations
4.
Which is the best estimate for the number of minutes on the calling card advertised below?
A 10 min
B 20 min
C 50 min
D 200 min
To estimate the number of minutes on the calling card, divide $10 by $0.05.
So, there are about 200 minutes on the calling card. Choice D is the correct answer.
GRIDDED RESPONSE The scale factor for two similar triangles is 2:3. The perimeter of the smaller triangle is 56
cm. What is the perimeter of the larger triangle in centimeters?
Use a proportion to find the perimeter of the larger triangle.
The perimeter of the larger triangle is 84 centimeters.
Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the
pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1
3
ft = 7.5 gal)
F 37.7 min
G7h
H 25.4 h
J 35.3 h
To find about how long it will take to drain the pool, first calculate the amount of water in the pool.
3
3
There are about 1413ft of water in the pool. Because 1 ft = 7.5 gallon, then
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The perimeter of the larger triangle is 84 centimeters.
Mr. Morrison is draining his cylindrical pool. The pool has a radius of 10 feet and a standard height of 4.5 feet. If the
pool water is pumped out at a constant rate of 5 gallons per minute, about how long will it take to drain the pool? (1
3
ft = 7.5 gal)
F 37.7 min
G7h
H 25.4 h
J 35.3 h
To find about how long it will take to drain the pool, first calculate the amount of water in the pool.
3
3
There are about 1413ft of water in the pool. Because 1 ft = 7.5 gallon, then
.
Use the equation t = w r, where t = time to drain the pool, w
r = rate water is
pumped to model the scenario. If the pool water is pumped out at a constant rate of 5 gallons per minute, it will take
minutes by 60 minutes which is 35.325
STATISTICS Look at the golf scores for the five players in the table.
Which of these is the range of the golf scores?
A 10
B 25
C 35
D 40
To find the range subtract the least score from the greatest score.
103 78 = 25
Choice B is the correct answer.
GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more
than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a
gallon of gas?
eSolutions
Let xManual
be the- Powered
numberbyofCognero
miles
Miles for a 4-
is
a luxury car travel on a gallon of gas.
10 miles
more
Miles for an 8-
Page 35
To find the range subtract the least score from the greatest score.
103 78 = 25
Choice B is the correct answer.
GAS MILEAGE A midsize car with a 4-cylinder engine travels 34 miles on a gallon of gas. This is 10 miles more
than a luxury car with an 8-cylinder engine travels on a gallon of gas. How many miles does a luxury car travel on a
gallon of gas?
Let x be the number of miles a luxury car travel on a gallon of gas.
Miles for a 4cylinder/ one
gallon
34
is
10 miles
more
than
=
10
+
Miles for an 8cylinder/one
gallon
X
A luxury car travel 24 miles on a gallon of gas.
DEER In a recent year, 1286 female deer were born in Clark County . That is 93 fewer than the number of male
deer born. How many male deer were born that year?
Let m = the number of male deer that were born.
The number of
female deer
is
1286
=
93 fewer than the
number of male deer
born.
m 93
1379 male deer were born that year.
Translate each equation into a verbal sentence.
f 15 = 6
f
A number f
minus
15
15
=
is
6
6.
7
7
=
to equal
20
20.
3h + 7 = 20
3h
Three
times a
number h
2
k + 18 = 54
+
is
increased
by
m
Page 36
k
2
+
18
=
54
m
3h
+
7
Three
is
7
timesMulti-Step
a
increased
2-3 Solving
Equations
number h
by
2
k + 18 = 54
m
2
+
and
added
to
k
A
number
k is
squared
3p = 8p
18
18
54
54
decreased
by
m
m.
=
is
the same as
8p r
the difference of
8
times p and r.
=t
+
=
t
added to
is
t.
t
Three
fifths of t
v=
=
to equal
20
20.
r
3p
Three multiplied
by a number p
t+
=
to equal
v+4
=
v
The product of
is equal
v
v
to
the product of
+
4
plus
4.
and v
GEOGRAPHY The Pacific Ocean covers about 46% of Earth. If P represents the surface area of the Pacific
Ocean and E represents the surface area of Earth, write an equation for this situation.
46% written as a decimal is 0.46.
Surface Area of the
= percent
Pacific Ocean
P
= 0.46
Surface Area of
the Earth
E
P = 0.46E.
Find the value of n in each equation. Then name the property that is used.
1.5 + n = 1.5
Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.
Page 37
Surface Area of the
= percent
Pacific Ocean
P
= 0.46
Surface Area of
the Earth
E
P = 0.46E.
Find the value of n in each equation. Then name the property that is used.
1.5 + n = 1.5
Because 1.5 + 0 = 1.5, n = 0. This is the Additive Identity.
8n = 1
Because 8
4
= 1, n =
. This is the Multiplicative Inverse.
n=0
Because 4
4 = 0, n = 4. This is the Additive Inverse.
1 = 2n
Because 1 = 2
,n =
. This is the Multiplicative Inverse.
Evaluate each expression.
2
5 + 3(4 )
3
[5(1 + 1) ]
[8(2)
2
4 ] + 7(4)
Page 38
[8(2)
2
4 ] + 7(4)
Page 39

Check - Waynesville R

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2. SOLUTION - Waynesville R