M8 – SWS – Investigation 2
Say It With Symbols
Investigation #2
Combining Expressions
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M8 – SWS – Investigation 2
Say It With Symbols
Investigation #2
Combining Expressions
Problem 2.1: What are the advantages and disadvantages of using one equation rather than
two or more equations to represent a situation?
Problem 2.2: What are some ways that you can combine one or more expressions (or
equations) to create a new expression (or equation)?
Problem 2.3: What equations represent the relationships among the volumes of cylinders,
cones, and spheres?
Problem 2.4: What formulas are useful in solving problems involving volumes of cylinders,
cones, and spheres?
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Equations with Like Terms
Solve and check:
1. 8y – 3y + 7 = 87
2.
5n + 3n – n + 5 = 26
3. 7d – 12 + 2d + 3 = 18
4.
7x – 2 – 12x = 63
Reflect
What steps are needed to solve an equation like the ones above?
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M8 – SWS – Investigation 2
Solve and Check
5. 5(x – 3) = 10
6.
-6(3x + 1) = -42
7. 5(y + 2) + 3 = 43
8.
27x – 3(x – 6) = 42
What are the steps needed to solve an equation like the ones above?
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M8 – SWS – Investigation 2
Define a variable, write and equation and solve. Clearly state your final answer.
8. A rectangular garden has a width of x feet and a length of x + 5 feet. If the
perimeter of the garden is 50 feet, find the length and width of the garden.
9. Twice the sum of a number a 4 is equal to 30. Find the number.
10. The length of each side of an equilateral triangle is increased by 5 inches, so
the new perimeter is now 60 inches. Write an solve an equation to find the
original length of the equilateral triangle.
11. Solve for x:
p(x + q) = r
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Solving Equations with CLT and Distribute Practice
Solve and check each of the following.
1. 44 = 4(x + 6)
2.
x + 4(x + 3) = 72
3. 2(y – 4) + 12 = 92
4.
2(y – 1) – y = -2
4y – 2(y – 5) = -2
6.
1
( x  8)  15  3
2
7. 3(x – 2) + 2(x + 1) = -14
8.
4 – 3(x + 2) – 2(x – 4) = 1
5.
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9. The length of a rectangle is 5 inches and the width is x + 7 inches. If the
perimeter of the rectangle is 100 inches, find the width of the rectangle.
10. The sides of a regular hexagon are all increased by 2 units. If the perimeter of
the new hexagon is 72 units, find the length of one side of the original hexagon.
11. Write a real world situation that be modeled by the equation 2(n + 20) = 110.
Then solve the equation.
12. Miguel and three of his friends went to the movies. They originally had a total of
$40. Each boy had the same amount of money and spent $7.50 on a ticket.
How much did each boy have left after buying his ticket?
a. Draw a bar diagram (tape diagram) to represent this situation.
b. Solve your equation.
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M8 – SWS – Investigation 2
Problem 2.1 ACE#1-5, 17-21, 40
1. The student council is organizing a T-shirt sale to raise money for a local charity.
They make the following estimates of expenses and income:
•
Expense of $250 for advertising
•
•
Expense of $4.25 for each T-shirt •
Income of $12 for each T-shirt
Income of $150 from a sponsor
a. Write an equation for the income 𝐼 for selling 𝑛 T-shirts.
b. Write an equation for the expenses 𝐸 for selling 𝑛 T-shirts.
c. Suppose the student council sells 100 T-shirts. What is their profit?
d. Write an equation for the profit 𝑃 made for selling 𝑛 T-shirts.
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M8 – SWS – Investigation 2
Problem 2.1 ACE#1-5, 17-21, 40
For Exercises 2–5, use the following
information: In Variables and Patterns,
several students were planning a bike tour.
They estimated the following expenses and
incomes.
2.
a. Write an equation for the total
expenses 𝐸 for 𝑛 bikers.
b. Write an equation for the total income 𝐼 for 𝑛 bikers.
c. Write an equation for the profit 𝑃 for 𝑛 bikers.
d. Find the profit for 25 bikers.
e. Suppose the profit is $1,055. How many bikers went on the trip?
f. Does the profit equation represent a linear, quadratic, or exponential function, or
none of these? Explain.
3. Multiple Choice Suppose someone donates a van at no charge. Which equation
represents the total expenses?
A.
𝐸 = 125 + 30
B.
𝐸 = 125𝑛 + 30𝑛
C.
𝐸 = 155
D.
𝐸 = 155 + 𝑛
4. Multiple Choice Suppose students use their own bikes. Which equation represents
the total expenses? (Assume they will rent a van.)
F.
𝐸 = 125𝑛 + 700
G.
𝐸 = 125 + 700 + 𝑛
H.
𝐸 = 825𝑛
J.
𝐸 = 350𝑛 + 125𝑛 + 700
5. Multiple Choice Suppose students use their own bikes. Which equation represents
the profit? (Assume they will rent a van.)
A.
𝑃 = 350 − (125 + 700 + 𝑛) B.
𝑃 = 350𝑛 − 125𝑛 + 700
C.
𝑃 = 350𝑛 − (125𝑛 + 700)
D.
𝑃 = 350 − 125𝑛 – 700
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Problem 2.1 ACE#1-5, 17-21, 40
17. Multiple Choice Which statement is false when 𝑎, 𝑏, and 𝑐 are different real
numbers?
F.
(𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐)
G.
𝑎𝑏 = 𝑏𝑎
H.
(𝑎𝑏)𝑐 = 𝑎(𝑏𝑐)
J.
𝑎 − 𝑏 = 𝑏– 𝑎
For Exercises 18–20, use the Distributive Property and sketch a rectangle to show
the equivalence of the two expressions.
18.
𝑥(𝑥 + 5) and 𝑥 2 + 5𝑥
19.
(2 + 𝑥)(2 + 3𝑥) and 4 + 8𝑥 + 3𝑥 2
20.
(𝑥 + 2)(2𝑥 + 3) and 2𝑥 2 + 7𝑥 + 6
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Problem 2.1 ACE#1-5, 17-21, 40
21. A student’s solution for 11𝑥 − 12 = 30 + 5𝑥 is shown below. Some steps are
missing in the solution.
𝟏𝟏𝒙 − 𝟏𝟐 = 𝟑𝟎 + 𝟓𝒙
𝟏𝟏𝒙 = 𝟒𝟐 + 𝟓𝒙
𝟔𝒙 = 𝟒𝟐
𝒙 = 𝟕
a. Fill in the missing steps above.
b. How can you check that 𝑥 = 7 is the correct solution?
c. Explain how you could use a graph or a table to solve the original equation for 𝑥.
40. The Phillips Concert Hall staff estimates their concession stand profits 𝑃𝐶 and
admission profits 𝑃𝐴 with the following equations, where 𝑥 is the number of people
attending (in hundreds):
𝑃𝐶 = 15𝑥 − 500
𝑃𝐴 = 106 𝑥 − 𝑥 2
The concession-stand profits include revenue from advertising and the sale of food
and souvenirs. The admission profits are based on the difference of ticket sales and
cost.
a. Write an equation for the total profit for 𝑃 in terms of the number of people 𝑥
(in hundreds).
b.
i. What is the maximum profit?
ii. How many people must attend in order to achieve the maximum
profit?
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M8 – SWS – Investigation 2
Problem 2.2 ACE#6-9, 22-31
For Exercises 6–8, recall the equations from Problem 2.2, 𝑷 = 𝟐. 𝟓𝟎𝑉 − 𝟓𝟎𝟎 and
𝑽 = 𝟔𝟎𝟎 − 𝟓𝟎𝟎𝑹.
6. Suppose the probability of rain is 50%. What profit can the concession stand expect to
make?
7. What is the probability of rain if the profit expected is $100?
8. The manager estimates the daily employee-bonus fund B (in dollars) from the number of
visitors V using the equation B = 100 + 0.50V.
a. Suppose the probability of rain is 30%. What is the daily employee-bonus fund?
b. Write an equation that relates the employee-bonus fund B to the probability of rain R.
c. Suppose the probability of rain is 50%. Use your equation to calculate the employeebonus fund.
d. Suppose the daily employee-bonus fund is $375. What is the probability of rain?
9. A park manager claims that the profit P for a concession stand depends on the number of
visitors V, and that the number of visitors depends on the day’s high temperature T (in
degrees Fahrenheit). The following equations represent the manager’s claims:
𝑃 = 4.25𝑉 – 300
𝑉 = 50(𝑇 − 45)
a. Suppose 1,000 people visit the park one day. Predict the high temperature on that day.
b. Write an expression for profit based on temperature.
c. Write an expression for profit that is equivalent to the one in part (b). Explain what
information the numbers and variables represent.
d. Find the profit if the temperature is 70°F.
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Problem 2.2 ACE#6-9, 22-31
22. In the following graph, line ℓ1
represents the income for selling
𝑛 soccer balls. Line ℓ2 represents
the expenses for manufacturing
𝑛 soccer balls.
a. What is the start-up expense (the expense before any soccer balls are produced) for
manufacturing the soccer balls? Note: The vertical axis is in thousands of dollars.
b. What are the expenses and income for producing and selling:
i. 500 balls
ii. 1,000 balls
iii. 3,000 balls
iv. Explain
c. What is the profit for producing and selling:
i. 500 balls
ii. 1,000 balls
iii. 3,000 balls
iv. Explain
d. What is the break-even point? Give the number of soccer balls and the expenses.
e. Write equations for the expenses, income, and profit. Explain what the numbers and
variables in each equation represent.
f.
Suppose the manufacturer produces and sells 1,750 soccer balls. Use the equations in
part (e) to find the profit.
g. Suppose the profit is $10,000. Use the equations in part (e) to find the number of soccer
balls produced and sold.
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M8 – SWS – Investigation 2
Problem 2.2 ACE#6-9, 22-31
For Exercises 23–28, use properties of equality to solve each equation. Check your
solution.
23.
7𝑥 + 15 = 12𝑥 + 5
24.
7𝑥 + 15 = 5 + 12𝑥
25.
−3𝑥 + 5 = 2𝑥 – 10
27.
9 − 4𝑥 =
3+𝑥
2
26.
14 − 3𝑥 = 1.5𝑥 + 5
28.
−3(𝑥 + 5) =
2𝑥−10
3
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Problem 2.2 ACE#6-9, 22-31
29. The writing club wants to publish a book of students’ short stories, poems, and
essays. A member of the club contacts two local printers to get bids on the cost of
printing the books.
a. Make a table of values (number of books printed, cost) for each bid. Use your table
to find the number of books for which the two bids are equal. Explain how you
found your answer.
b. Make a graph of the two equations. Use your graph to find the number of books
for which the two bids are equal. Explain.
c. For what numbers of books is Bid 1 less than Bid 2? Explain.
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Problem 2.2 ACE#6-9, 22-31
30. Use the information about printing costs from Exercise 29.
a. For each bid, find the cost of printing 75 books.
b. Suppose the cost cannot exceed $300. For each bid, find the greatest number of
books that can be printed. Explain.
31. The writing club decides to request bids from two more printers.
For what number of books does Bid 3 equal Bid 4? Explain.
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M8 – SWS – Investigation 2
Problem 2.3 ACE#10-12, 32-34, 41-43
10. Explain what happens to the volume of a cylinder 𝑉 = 𝜋𝑟 2 ℎ when
a. the radius is doubled.
b. the height is doubled.
1
11. Explain what happens to the volume of a cone when 𝑉 = 𝜋𝑟 2 ℎ
3
a. the radius is doubled.
b. the height is doubled.
4
12. Explain what happens to the volume of a sphere when 𝑉 = 𝜋𝑟 3
3
a. the radius is doubled.
b. the radius is tripled.
c. the radius is quadrupled (multiplied by 4).
Simplify each expression. Then explain how the expression could represent a volume
calculation. (3.14 is used as an approximation of 𝝅)
32.
3.14 × 4.252 × 5.5
33.
1
34.
3
4
3
× 3.14 × 4.252 × 5.5
× 3.14 × 4.253
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Problem 2.3 ACE#10-12, 32-34, 41-43
For Exercises 41–43, suppose you slice the three-dimensional figure as indicated.
Describe the shape of the face that results from the slice.
41.
a.
vertical slice
b.
horizontal slice
42.
a.
vertical slice
b.
horizontal slice
43.
a.
vertical slice
b.
horizontal slice
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Problem 2.4 ACE#13-16, 35-39
13. The astronomy observatory
pictured below has a diameter of 10
feet.
a. What is the area of the floor?
b. Write a general algebraic expression for the area of the floor.
c. The observatory is made of a 3 foot-tall cylinder and half of a sphere (also called
a hemisphere). What is the volume of the space inside the observatory?
d. Write a general algebraic expression for the volume of the space inside the
observatory.
14. The Jackson Middle School model rocket club drew the model rocket design below.
The rocket is made from a cylinder and a cone. Write an expression to represent the
volume of the rocket.
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Problem 2.4 ACE#13-16, 35-39
15. Ted made the model submarine shown below for his science class. Write an
algebraic expression for the volume of Ted’s submarine.
16. The pyramid and rectangular prism have the same
base and height.
a. Find the volume of the pyramid.
1
b. Draw a pyramid with a volume of 3 (8) cubic units. Hint: You might find it easier
to draw the related prism first.
1
c. Draw a pyramid with a volume of 3 (27) cubic units.
d. Find the height of a pyramid with a volume of 9𝑥 3 cubic units.
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Problem 2.4 ACE#13-16, 35-39
Simplify each expression. Then explain how the expression could represent a volume
calculation. (3.14 is used as an approximation of 𝝅)
35.
1
4
36.
3.14 × 4.252 × 5.5 + 6 × 3.14 × 4.253
37.
3.14 × 4.252 × 5.5 + 3 × 3.14 × 4.252 × 3.5
38.
3.14 × 4.252 × 5.5 − × 3.14 × 4.253
39.
3.14 × 4.252 × 5.5 − 3 × 3.14 × 4.252 × 3.5
× 3.14 × 4.252 × 5.5 + 6 × 3.14 × 4.253
3
4
1
4
6
1
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Summary
Investigation #2
Combining Expressions
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