Name ____________________________________________ Date ____________ Class ____________
Investigation 1
1ACE Exercise 17
The Shapes of Algebra
17. The circle in this design is centered at the origin.
y S(0, 6)
a. Find coordinates for points J, K, and L on the circle.
P
R
J:
x
L
K:
J
T
L:
V
K
b. Points P, R, V, and T are midpoints of the segments on
which they lie.
a. Find coordinates for each of these points.
P:
R:
HINT Midpoint is the point halfway
between the two end points of a line
segment. Thus P is halfway between
L and S.
V:
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T:
c. Find coordinates of the vertices of the innermost quadrilateral.
c. Is this quadrilateral a square?
HINT
What is special about a square?
Explain.
205
Name ____________________________________________ Date ____________ Class ____________
Investigation 2
2ACE Exercise 1
The Shapes of Algebra
1. a. Sam needs to rent a car for his one-week trip in Oregon. He is considering
two companies. A+ Auto Rental charges $175 plus $0.10 per mile. Zippy
Auto Rental charges $220 plus $0.05 per mile. Write an equation relating
the rental car cost C for each company to the miles m driven.
1. a. Zippy Auto Rentals: C =
1. a. Cruise the Country: C =
b. Graph the equations.
Miles
206
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Cost
Auto Rental Charges
Name ____________________________________________ Date ____________ Class ____________
Investigation 2
2ACE Exercise 1 (continued)
The Shapes of Algebra
Use your graph from part (b) to answer parts (c) – (d).
c. Under what circumstances is the rental cost the same for both companies?
c. What is that cost?
d. Under what circumstances is renting from Zippy cheaper than renting
from A+?
When is the cost of renting from
Zippy lower than the cost of renting
from A+?
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HINT
e. Suppose Sam rents a car from A+ and drives it 225 miles. What is his
rental cost?
207
Name ____________________________________________ Date ____________ Class ____________
3ACE Exercise 2
Investigation 3
The Shapes of Algebra
2. Neema saves her quarters and dimes. She plans to exchange the coins for
paper money when the total value equals $10.
a. How many coins does she need to make $10:
a. if all the coins are quarters ($0.25)?
How many quarters are in $1?
How many are in $10?
a. if all the coins are dimes ($0.10)?
How many dimes are in $1?
How many are in $10?
208
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b. What equation relates the number of quarters x and the
number of dimes y to the goal of collecting $10?
Name ____________________________________________ Date ____________ Class ____________
Investigation 3
3ACE Exercise 2 (continued)
The Shapes of Algebra
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c. Use the answers from part (a) to draw a graph showing all solutions to the
equation in part (b).
d. Use the graph from part (c) to find five other combinations of dimes and
quarters (y and x) that will allow Neema to reach her goal.
Example: 20 quarters and 50 dimes would equal $10
20 quarters = .25 ⫻ 20 = $5
50 dimes = .10 ⫻ 50 = $5
Together that equals $10
209
Name ____________________________________________ Date ____________ Class ____________
Investigation 4
4ACE Exercises 2–7
The Shapes of Algebra
2. Solve each system.
a. y = 6x + 4
y = 4x – 2
6x + 4 = 4x – 2
In this system of equations, y is set
equal to two different things. Since y is
equal to both of them, and y is the same in
both equations, set the two different
expressions equal.
HINT
To get “like terms” on the same side, subtract 4x from both sides
of the equation.
6x + 4 – 4x = 4x – 2 – 4x
Simplify.
2x + 4 = –2
Subtract 4 from both sides.
2x + 4 – 4 = –2 – 4
2x = –6
Divide both sides by 2.
2x = ⫺6
2
2
Simplify.
x = –3
To check, substitute the value of x (–3) into both equations in the system
and simplify.
y = 6(–3) + 4
y = (–18) + 4
y = –14
y = 4(–3) – 2
y = –12 – 2
y = –14
Both equations have the same value for y. So in this system of equations,
x = –3 and y = –14.
210
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Simplify.
Name ____________________________________________ Date ____________ Class ____________
Investigation 4
4ACE Exercises 2–7 (continued)
The Shapes of Algebra
Use the example on the previous page to help you solve Exercises 3–7.
3. y = 3x + 7
y = 5x – 7
5. y = –x + 16
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y = 7x – 8
4. y = –2x – 9
y = 12x + 19
6. y = 17x – 6
y = 12x + 44
7. y = –20x + 14
y = – 8x – 44
211
Name ____________________________________________ Date ____________ Class ____________
Investigation 5
5ACE Exercise 1
The Shapes of Algebra
1. Ana has a car and a motorcycle. She wants to limit the combined mileage of
the two vehicles to total at most 500 miles per month.
a. Write an inequality to model this condition.
Let c = the number of miles the
car is driven and let m = the number of
miles the motorcycle is driven.
HINT
b. Draw a graph of all the (car miles, motorcycle miles) pairs that satisfy this
condition.
HINT Car miles = c and
motorcycle miles = m
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c. What strategy did you use to draw your graph?
212
Name ____________________________________________ Date ____________ Class ____________
Unit Test
The Shapes of Algebra
This diagram shows a quadrilateral with vertices on a circle of radius 5
and center (0, 0) on a coordinate grid.
y
A
D
O
B
x
C
1. Write an equation satisfied by the (x, y) coordinates of points on
the circle.
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2. Is side AD parallel to side BC? Explain how you know.
3. Is side AB perpendicular to CB? Explain how you know.
4. Write an equation for a line that is parallel to side AD and passes
through the point (0, 1).
213
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
The Shapes of Algebra
José is batboy for the Lansing Lugnuts baseball team during summer vacation.
His pay includes two season tickets worth a total of $80 and $5 per hour of
practice or game time that he works.
Charlie works at the concession stand for most Lugnut games, earning $6 per
hour plus a $50 bonus if he works at least 20 games.
5. Write an equation showing how José’s summer pay P depends on the number
of hours he works h.
HINT Remember, Jose gets $80 worth of
season tickets no matter how many games
he works.
P=
6. Write an inequality that answers the question, “For how many hours of work
will José’s total summer pay be less than $260?”
Solve the inequality.
8. Write an inequality to find the number of hours worked for which
Charlie will earn more in total summer pay than José.
Solve the inequality.
214
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7. Suppose that Charlie plans to work at least 20 games. Write an equation
showing how his total summer earnings E are related to the number of
hours he works n.
HINT Remember, if Charlie works at
least 20 games, he gets a bonus of $50.
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
The Shapes of Algebra
The Plano Texans are a youth drum and bugle corps that competes with music
and precision marching against other groups all over the country. The corps rents
instruments to members. Each bugle rents for $10 per month and each drum
rents for $5 per month.
9. What is the corps monthly income from instrument rentals if members rent:
a. 7 bugles and 9 drums?
HINT Write and solve an
equation for income I dependent
on bugles b and drums d.
b. 9 bugles and 7 drums?
10. What equation relates the number of bugle rentals b and the number of drum
rentals d to the business manager’s goal of $100 in monthly rental income?
11. Draw a graph showing solutions of the rental income equation on the
following grid and give coordinates of 3 points that represent solutions.
y
Coordinates
Number of Drums
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Instrument Rentals for $100
1.
2.
3.
x
Number of Bugles
215
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
The Shapes of Algebra
12. Suppose that there are 12 members of the drum and bugle corps who rent an
instrument.
a. What equation relating b and d expresses this fact?
b. Give two solutions to the equation in part (a).
c. Write and solve a system of equations that finds numbers of bugle rentals
b and drum rentals d that supply 12 members for a rental income of $100.
13. Suppose there are at most 18 members of the drum and bugle corps who
rent an instrument.
a. Write an inequality relating x and y that express this fact, keeping in mind
that in this context, x ⱖ 0 and y ⱖ 0.
Instrument Rentals for $100
Number of Drums
y
x
Number of Bugles
216
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b. Draw a graph illustrating the solutions to the inequality you wrote in
part (a).
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
The Shapes of Algebra
c. Suppose that the group would like a rental income of at least $100, but
they will rent at most 18 instrument rentals. Write a system of inequalities
to express these relationships.
d. Draw a graph exhibiting the possible numbers of drum and bugle rentals
in part (c).
Instrument Rentals for $100
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Number of Drums
y
x
Number of Bugles
e. Use your graph from part (d) to list two possibilities for numbers of drum
and bugle rentals which satisfy the conditions of part (c).
Possibility 1:
Possibility 2:
217
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
The Shapes of Algebra
For Exercises 14 and 15, solve the systems of linear equations so that your
solutions illustrate 3 different methods:
a. substitution
b. linear combinations
c. another strategy of your choosing
14.
2x + y = 7
x + 3y = 11
a.
c.
y = 4x – 3
3x + 2y = 16
a.
218
b.
c.
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15.
b.