4-9
Slopes of Parallel and
Perpendicular Lines
Focus on Modeling
Essential question: How can you use linear equations to model the results of a
fundraiser?
TEACH
Standards for
Mathematical Content
1
The following standards are addressed in this
lesson. (An asterisk indicates that a standard is
also a Modeling standard.) For more detailed
information, see the correlation for each section of
the lesson.
Write a linear equation to show the
amount of money raised.
Standards
N-Q.1.2 Define appropriate quantities for the
purpose of descriptive modeling.*
A-SSE.1.1 Interpret expressions that represent a
quantity in terms of its context.*
Number and Quantity: N-Q.1.1*,
N-Q.1.2*
Algebra: A-SSE.1.1*, A-SSE.1.1a*, A-CED.1.2*,
A-CED.1.3*
A-SSE.1.1a Interpret parts of an expression, such
as terms ... and coefficients.*
A-CED.1.2 Create equations in two . . . variables to
represent relationships between quantities . . . *
Prerequisites
• Graphing a linear equation
A-CED.1.3 Represent constraints by equations . . . *
Math Background
Questioning Strategies
• How can you find the solutions to the linear
equation representing the sales goal,
10x + 25y = 2000? Choose values for x and solve
In this lesson, students write a linear equation
to show the total amount of money that can be
raised by selling T-shirts for $10 each and blanket
wraps for $25 each. Students also use the graph
of a linear equation to show the sales required
to raise the minimum of $1000. In both cases,
the variables are constrained to positive x- and
y-values. Standard CC.9-12.A.CED.3 uses the term
constraint in a mathematical way, that students
may not be familiar with. Constraints are used in
college- and graduate-level mathematics, as well
as in more advanced high school mathematics
courses. This lesson is the first step in learning to
work with constraints.
for y or choose values of y and solve for x.
• How can you use your linear equation for the
sales goal to find the amount of money earned
by a combination of T-shirts and blanket wraps?
How can you find the money earned by selling
100 T-shirts and 100 blanket wraps? Evaluate
the expression 10x + 25y on the left side of the
equation; when x = 100 and y = 100,
10(100) + 25(100) = 3500, or $3500.
INTR O D U C E
• Suppose you earn exactly $2500 by selling a
combination of T-shirts and blanket wraps.
Describe how to find how many blanket wraps
were sold if you know how many T-shirts were
sold. Use an example. Substitute the known
value of x into 10x + 25y = 2500, and then solve
for y. For example, when x = 50, 10(50) + 25y =
2500, or 25y = 2000. Therefore y = 100. You sold
Ask students how they can calculate the amount
of money they can raise if they sell one item for a
certain price and another item for another price.
Ask students to discuss how they would find out
how many of each item to sell to break even and
then go on to meet a sales goal. Tell students that
these questions can be answered by finding the
solutions to linear equations.
100 blanket wraps.
continued
Chapter 4 241
Lesson 9
© Houghton Mifflin Harcourt Publishing Company
• Determine whether selling 200 T-shirts alone will
meet the goal. How do you know? Yes;
10(200) + 25(0) = 2000
Name
Class
Notes
4-9
Date
Slopes of Parallel and Perpendicular
Lines
Focus on Modeling
N-Q.1.1*,
N-Q.1.2*,
A-SSE.1.1*,
A-SSE.1.1a*,
A-CED.1.2*,
A-CED.1.3*
Essential question: How can you use linear equations to model the results
of a fundraiser?
T
he Band Booster Club is selling T-shirts and blanket wraps to raise money for
a trip. The band director has asked the club to raise at least $1000 in sales. So,
the booster club has set a fundraising goal of $2000 in sales.
T
he booster club president wants to know how many T-shirts and how many
blankets wraps the club needs to sell to meet their goal of $2000. The T-shirts
cost $10 each and the blanket wraps cost $25 each. How can the booster club
president use the sales price of each item to help the fundraiser meet its goal?
1
Write a linear equation to show the amount of money raised.
A
Let x equal the number of T-shirts sold. Write an expression for the amount
of money raised from T-shirt sales. Interpret the expression.
B
Let y equal the number of blanket wraps sold. Write an expression for the amount
of money raised from blanket wrap sales. Interpret the expression.
C
Combine the expressions and the sales goal in a linear equation.
10x; the booster club will get $10 for each of x T-shirts sold.
© Houghton Mifflin Harcourt Publishing Company
25y; the booster club will get $25 for each of y blanket wraps sold.
10x + 25y = 2000
REFLECT
1a. Determine whether the booster club will meet their goal if they sell 50 T-shirts and
50 blanket wraps. Explain.
No; 10(50) + 25(50) = 1750 and $1750 < $2000.
1b. Determine whether the booster club will meet their goal if they sell 100 T-shirts and
40 blanket wraps. Explain.
Yes; 10(100) + 25(40) = 2000 and $2000 = $2000.
241
Chapter 4
Lesson 9
A1_MFLBSE202160_C04L09.indd 241
2
Graph the linear equation.
A
B
Calculate three pairs of values for x and y. Enter your results in the table.
x
y
0
80
200
0
50
60
Plot the ordered pairs on the coordinate grid. Connect the points to graph
the equation.
Number of blanket wraps sold
100
y
90
80
10x + 25y = 2000
70
(50, 60)
60
50
40
30
20
10
x
0
20
40
60
80
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
03/05/14 4:40 PM
100 120 140 160 180 200 220
Number of T-shirts sold
C
Label the x- and y-axes in the boxes provided.
REFLECT
2a. What does the point where the line intersects the x-axis represent?
The point (200, 0) represents the fact that the booster club must sell
200 T-shirts to reach their goal of $2000 if they sell no blanket wraps.
Chapter 4
A1_MNLESE882000_C04L09.indd 242
Chapter 4 242
Lesson 9
3/10/12 4:54:31 PM
242
Lesson 9
1
Avoid Common Errors
Point out to students that the scales on the
x-axis and y-axis are not the same for any of the
coordinate grids provided in this lesson. Students
will need to carefully match the appropriate
variable with its axis and to make sure that each
point is plotted correctly.
continued
Teaching Strategy
If students have difficulty answering Reflect
Questions 1a and 1b, suggest that they make a table
with three columns: x, y, and 10x + 25y. Test various
values of x and y by using multiples of 10. Those
rows that have 10x + 25y = 2000 are solutions of the
linear equation. Have students adjust their values
for x and y to try to achieve a value of 2000 in the
third column.
2
Technology
Students can graph equations on a graphing
calculator to check their graphs. First, students
will have to solve the equation for y. By using the
Y= key, they can enter an equation as an
expression. Finally, they press the GRAPH key.
Graph the linear equation.
Standards
N-Q.1.1 . . . choose and interpret the scale and the
origin in graphs . . .*
Students will need to adjust the window settings
carefully if they want their graphs to match the
coordinate planes in the lesson.
A-CED.1.2 . . . graph equations on coordinate axes
with labels and scales.*
3
A-CED.1.3 . . . interpret solutions as viable or
nonviable options in a modeling context.*
Write and graph a linear equation
that shows the sales required to raise
the minimum of $1000.
Standards
N-Q.1.1 . . . choose and interpret the scale and the
origin in graphs . . .*
Questioning Strategies
• What do points on the graph of 10x + 25y = 2000
represent? the combination of T-shirts for $10
A-CED.1.3 Represent constraints by
equations, ... and interpret solutions as viable or
nonviable options in a modeling context.*
and blanket wraps for $25 that must be sold to
raise $2000
• Do all the points on the line represent solutions
in the context of the problem? Why or why not?
Questioning Strategies
• Why are the intercepts of the line important to
find? They are endpoints of the line since the
variables must be nonnegative. They are also the
simplest points to use to graph the line.
• How will the progressive sales of T-shirts affect
the number of blanket wraps that need to be sold,
or vice versa? The more you sell of one item, the
less you need to sell of the other item.
• Which values of x and y will result in a need to do
another fundraiser to make the minimum goal of
$1000? How do you know? Any values of x and
y that result in 10x + 25y being less than 1000
would necessitate another fundraiser because
they would represent not making a total of
$1000. These are the points in the first quadrant
below the line 10x + 25y = 1000.
Chapter 4
243
Lesson 9
© Houghton Mifflin Harcourt Publishing Company
No; you cannot sell fractional parts of T-shirts or
blanket wraps. While all points on the line are
solutions of 10x + 25y = 2000 in a mathematical
sense, only the points that have integer values
for both x and y are solutions of the problem in
a real-world sense.
Notes
2b. Explain what the point (100, 40) represents on this graph.
The point (100, 40) represents the fact that if the booster club sells 100
T-shirts, they must sell 40 blanket wraps to meet their goal of $2000.
2c. Suppose the Band Booster Club sells 25 blanket wraps during a chilly
football game. Use the graph to determine about how many T-shirts
they need to sell.
about 140 T-shirts
3
W
Write
and graph a linear equation that shows the sales required
tto raise the minimum of $1000.
A
Use the expressions for the amount of money raised from T-shirts
and blanket wraps and the minimum goal of $1000 to write a
linear equation.
B
Calculate three pairs of values for x and y.
Enter your results in the table. Then, graph
the linear equation.
10x + 25y = 1000
y
0
40
100
0
50
20
y
90
Number of blanket wraps sold
© Houghton Mifflin Harcourt Publishing Company
100
x
80
70
60
50
40
30
(50, 20)
20
10
0
10x + 25y = 1000
20
40
x
60
80
100 120 140 160 180 200 220
Number of T-shirts sold
Chapter 4
C
Graph the linear equation you found in part 1 that represents the sales goal of
$2000 on the same axes.
Number of blanket wraps sold
100
y
90
80
70
60
50
40
30
10x + 25y = 2000
(50, 20)
20
10x + 25y = 1000
10
x
0
20
40
60
80
100 120 140 160 180 200 220
Number of T-shirts sold
REFLECT
3a. Make a conjecture about the graphs of the two linear equations. How could you show
your conjecture is true?
The graphs are parallel. Find the slope of each line and show that they are equal.
Both slopes are -0.4, so the lines are parallel.
3b. Suppose the manufacturer made an error and none of the blanket wraps were
sellable. How many T-shirts would the Band Booster Club need to sell
to raise $1000? to raise $2000?
100 T-shirts; 200 T-shirts
3c. If the booster club sells 18 blanket wraps, what is the number of T-shirts they need to
sell to raise $1000? to raise $2000?
The first equation would be 10x + 25(18) = 1000, or 10x = 550. Dividing both
sides by 10 gives x = 55. They would need to sell at least 55 T-shirts. The second
equation would be 10x + 25(18) = 2000, or 10x = 1550. Dividing both sides by
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Lesson 9
243
10 gives x = 155. They would need to sell 155 T-shirts.
3d. Based on your answers to questions 3b and 3c, how many more T-shirts must the
Booster Club sell for a given number of blanket wraps sold to increase their total
from $1000 to $2000?
100 more T-shirts
Chapter 4
Chapter 4
244
Lesson 9
244
Lesson 9
CLOSE
Highlighting the
Standards
Essential Question
How can you use linear equations to model the
results of a fundraiser?
Linear equations can be used to model the total
amount raised from selling two different items.
Any ordered pair that is a solution of the linear
equation represents a situation under which the
goals of the fundraiser have been met.
The steps in this lesson address Mathematical
Practice Standard 5 (Model with
mathematics). Discuss with students how the
sales figures for T-shirts and blanket wraps
are ordered pairs that can be graphed. Show
them how the ordered pairs that meet a
particular sales goal lie on a line.
Summarize
Have students write a journal entry in which they
provide a one-page summary of examples of the
number of T-shirts and blanket wraps that could
be sold in order to meet the minimum sales goal of
$1000 and the desired sales goal of $2000.
E X TE N D
Standards
N-Q.1.1 Use units as a way to understand
problems and to guide the solution of multi-step
problems . . . * (Ex. 1)
N-Q.1.2 Define appropriate quantities for the
purpose of descriptive modeling.* (Exs. 3, 5)
A-CED.1.2 . . . graph equations on coordinate axes
with labels and scales.* (Ex. 2)
A-CED.1.3 Represent constraints by
equations, . . . and interpret solutions as viable
or nonviable options in a modeling context.*
(Exs. 3, 4, 5, 6)
© Houghton Mifflin Harcourt Publishing Company
Chapter 4
245
Lesson 9
Notes
EXTEND
1. Use unit analysis to determine the unit of measurement for the expression 10x + 25y.
dollars
dollars
_____
· T-shirts + _____
· wraps = dollars + dollars = dollars
wrap
T-shirt
2. Describe where the lines were in relation to each other when you graphed
them both on the same coordinate grid.
The line for the linear equation 10x + 25y = 2000 is above the line for the linear
equation 10x + 25y = 1000 and is parallel to it.
3. The booster club sold 30 more T-shirts than blanket wraps at a fundraising event
and made exactly $1000. How many of each did they sell?
Let b equal the number of blanket wraps sold. So, the money raised by the
blanket wraps is 25b and the money raised by T-shirts is 10(b + 30).
10(b + 30) + 25b = 1000
10b + 300 + 25b = 1000
-300
-300
+ 25b = 700
10b
35b = 700
35b
700
___
= ___
© Houghton Mifflin Harcourt Publishing Company
35
35
b = 20
The booster club sold 20 blanket wraps and 20 + 30 = 50 T-shirts.
4. If the booster club orders 100 blanket wraps, they get a reduced price on them. Does
it make sense for the booster club to order 100 blanket wraps if the goal is to raise
$2000? Explain your answer.
Sample answer : No. Even if the club sold no T-shirts at all, the number of
blanket wraps they need to sell to make their goal is 80. Unless the savings is
significant, the booster club does not need to order 100 blanket wraps.
Chapter 4
245
Lesson 9
5. The booster club will earn $5 profit for every T-shirt sold and $10 profit for every
blanket wrap sold. Write an equation to show how many T-shirts and blanket wraps
the club would need to sell to make a profit of $1000.
Let x equal the number of T-shirts sold and y equal the number of blanket
5x + 10y = 1000
6. Find a solution to the linear equation you wrote in Problem 5. How much money
would the booster club raise in sales if they sold that many T-shirts and blanket wraps?
Sample answer : Let x = 40.
5(40) + 10y = 1000
200 + 10y = 1000
- 200
- 200
10y = 800
y = 80
So, if the booster club sells 40 T-shirts and 80 blanket wraps, the club will
earn a $1000 profit.
10(40) + 25(80) = sales
400 + 2000 = 2400
The booster club would raise $2400 in sales to earn a $1000 profit.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
wraps sold.
Chapter 4
Chapter 4
246
Lesson 9
246
Lesson 9
ADD I T I O NA L P R AC TI C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. y = 3x + 4 and y = 3x; y = 4 and y = 3
1 x + 4 and y = _
1 x+1
2. y = _
2
2
2
2
_
_
3. - ; undefined; - ; undefined;
3
3
The opposite sides have the same slope
which means they are parallel.
A quadrilateral is a parallelogram if the
opposite sides are parallel.
4. 10x + 20y = 2000
5. 10x + 20y = 2000
6. Yes; the graphs have the same slope.
7. For 10x + 20y = 2000, the y-intercept is 100.
For 10x + 20y = 4000, the y-intercept is 200.
The number of sweatshirts that must be sold
in each case to raise either $2000 or $4000.
© Houghton Mifflin Harcourt Publishing Company
8. For 10x + 20y = 2000, the x-intercept is 200.
For 10x + 20y = 4000, the x-intercept is 400.
The number of T-shirts that must be sold in
each case to raise either $2000 or $4000.
Problem Solving
1. The top and bottom are parallel because
they are both horizontal. The sides are parallel
5 . It is a
because they both have a slope of - _
3
parallelogram because both pairs of opposite
sides are parallel.
2. 5x + 10y = 500; 5x + 10y = 1000; They are
parallel because the graphs of the lines have
the same slope.
3. D
4. G
5. D
Chapter 4
247
Lesson 9
Name
Class
Notes
4-9
Date
Additional Practice
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Lesson 9
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Lesson 9