lesson 1: the race exercises

Copyright © 2014 Pearson Education, Inc. 5

Grade 8 Unit 5: Linear Equations

EXERCISES

EXERCISES

• Review your Unit Assessment from the previous unit.

• Write your wonderings about linear equations.

• Write a goal stating what you plan to accomplish in this unit.

• Based on your previous work, write three things you will do differently during this unit to increase your success.

LESSON 1: THE RACE



EXERCISES

EXERCISES

1. Solve. x + 3 = 2x – 1

Solve each equation.

2. 5n + 4 = 3(n – 2)

3. 2n + 3n – 8 = 4n + 7

4. 3(n – 4) = 5(n – 5)

5. 2n + 6 + n = 4n – 3n + 4

Solve.

6. Six times a number is equal to ten less than the sum of the number and 3. What is the number?

Challenge Problem

7. Write an equation that has a variable on both sides and a solution of 5.6. Write a word problem that matches your equation.

LESSON 2: ONE VARIABLE ON BOTH SIDES



EXERCISES

EXERCISES

1. Solve. x + 5 = 4(x – 1)

Write whether the equation has one solution, no solution, or all numbers as solutions.

2. 3x + 4 = 6 + 3x

3. x + 9 + 3x = 4x + 9

4. 3x + 4(x – 5) = 5x + 8

5. 3(x + 2) = 3x + 2

6. 2(x – 6) = –10 + 2x – 2

7. 5x + x + 4(x – 1) = 3x – 4 + x

Challenge Problem

8. a. Write an equation that has no solution.

b. Write an equation that has only 0 as a solution.

LESSON 3: HOW MANY SOLUTIONS?



EXERCISESLESSON 4: PARALLEL OR INTERSECTING?

EXERCISES

1. Write the expression 4(x + 5) + 4x + 8 – 2 in simplest form.

Create graphs of each expression in order to decide if the equation has one solution, no solution, or all numbers as solutions.

2. 4x + 6 = 4x – 6

3. 3(x + 4) = 3x + 7

4. 4x + 9 = 2 + 2(x + 1) + 2x + 5

5. 3x + 8 = 8(x + 1)

6. 3x – 4x – 1 = 7x – 6

7. 12

x + 6 = 3x – 1

Challenge Problem

8. Graph y = 2x + 5 and y = 3x + 1 on the same coordinate grid. Explain how to use the graph to find the solution to 2x + 5 = 3x + 1.



EXERCISES

EXERCISES

1. Which equation is equivalent to y = 4x + 5?

A 4x + y = 5 B 4x – y = –5

C 4x + y = –5 D 4x – y = 5

2. Write each equation in slope-intercept form: (y = mx + b)

a. x – y = 3b. 2x + 2y = 10c. –5x + 5y = 35d. x – y = 0

3. Write each equation in standard form: (ax + by = c)

a. y = 3x – 4

b. y = –12

x + 1

c. y = –2x + 2

4. What are the slope and y-intercept of each equation?

a. 2x + y = 8b. x – y = 4

LESSON 5: FORMS OF LINEAR EQUATIONS



EXERCISES

5. Look at the graph of this line:

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10

–5–4–3–2–1 10987654321–2–3–4–5–6 x

y

Write an equation in both slope-intercept form and standard form for the line.

6. Using the addition and multiplication properties of equality, show that the equation

10x + 35y = 70 is equivalent to y = − 27

x + 2.

Challenge Problem

7. Marshall likes to use interval training. He jogs at 200 meters per minute and runs 250 meters per minute. He runs 8 km every day. Write an equation in standard form. Let x represent the number of minutes of jogging and y represent the number of minutes of running for Marshall’s interval training.

LESSON 5: FORMS OF LINEAR EQUATIONS



EXERCISESLESSON 6: A SYSTEM OF EQUATIONS

EXERCISES

1. Write the equation 6x + 3y = 12 in slope intercept form. (y = mx + b)

Solve each system of equations. If the system has no solution, write no solution. If a system has infinitely many solutions write infinitely many solutions.

2. y = 2x + 3

y = 3x – 4

3. y = 3x – 2x

y – x = 4

4. 4 + y = 4x

y = 4(x – 1)

5. 3 = 3x + y

– 3x + y = 8

6. 9 = 5x – y

–3x + y = 9

Challenge Problem

7. Write a system of two equations that has no solution. Describe what the graph of the system would look like.



EXERCISESLESSON 7: SOLVING—SUBSTITUTION

EXERCISES

1. Solve this equation.

3(2y – 1) + 5y = 8

Solve each system of equations using the substitution method. If the system has no solution, write no solution. If a system has infinitely many solutions, write infinitely many solutions.

2. x = 4y + 2

2x – 3y = 9

3. y = 2x

3x + y = 12

4. 3x + y = 5

4x + 2y = 20

5. 4x + 5 = y

4x – y = 8

6. y = 3x – 7 + x

7 = 4x – y

Challenge Problem

7. Write a system of equations that has (4, 5) as the solution.



EXERCISES

EXERCISES

1. Add the two equations.

3x – y = 7

2x + y = –6

Solve each system of equations using the elimination method.

2. 4x – y = 9

3x + y = 5

3. x + 3y = 8

–x + 5y = 4

4. 5x + 3y = 8

4x + 3y = 5

5. 2x + 8y = 6

3x – 4y = –1

Challenge Problem

6. a. Explain why the substitution method, rather than the elimination method, might be the better choice for solving this system of equations.

x = 4y + 1

3y + 5x = 10

b. Solve the system of equations.

LESSON 8: SOLVING—ELIMINATION



EXERCISES

EXERCISES

1. The sum of two numbers is 25. Their difference is 9. What are the two numbers? Use a system of equations to find the two numbers.

2. A fire lookout is a person who looks for fire from the top of a structure known as a fire lookout tower. These towers are used in remote areas, often on mountain tops. From these towers, the fire lookouts have a good view of the surrounding terrain and are able to spot wildfires.

The grid shows the location of two lookout towers.

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–5–4–3–2

10987654321 x

y

Tower 1

Tower 2

A fire breaks out. The fire lookout at Tower 1 sees the fire with a line-of-sight having

slope of – . The fire lookout at Tower 2 sees the fire with a line-of-sight having

slope of 1.

At the coordination center, exact coordinates of the fire are needed in order to find the optimal position for a water-bombing plane.

a. What is the y-intercept and slope of each line? b. Find the linear equations of the two lines. c. Find the coordinates of the fire.

12

LESSON 9: USING A SYSTEM OF EQUATIONS



EXERCISES

Challenge Problem

3. A third tower is located at (5, 0). The lookout sees a fire with a line-of-sight having a

slope of –32

. Can this be the same location as the fire that Tower 1 and Tower 2 see?

Explain.

LESSON 9: USING A SYSTEM OF EQUATIONS



EXERCISES

EXERCISES

1. Write the equation in slope-intercept form for a line with slope 2 that passes through the point (2, 5).

2. Each of these tables represents a linear relationship.

Table A

x –2 –1 0 2

y –4 –2 0 4

Table B

x –4 –2 0 4

y –3 0 3 9

a. Find the slope of each line. b. Find the equation for each line. c. Find the coordinates of the point of intersection.

LESSON 10: SOLVING PROBLEMS



EXERCISESLESSON 10: SOLVING PROBLEMS

3. Jones and Janes are two competing landscaping companies.

Jones charges $40 to come to the location and then $40 per hour.

Janes made a graph representing her company’s costs.

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543 82 71 6 x

y

Hours

Dollars

Two points on the graph are (0, 60) and (3, 150).

The horizontal axis shows the number of hours. The vertical axis shows the total cost.

a. Which company is less expensive for a 3-hour job? b. Draw the graph of Jones in the same coordinate system.c. What are the coordinates of the point of intersection of the two lines? d. What is the meaning of this point?



EXERCISES

Challenge Problem

4. Points A, B, C, and D represent the locations of four airports.

1

2

3

4

5

–2

–154321–1–2–3–4 x

y

D

A

C

B

a. Find the slope of each line segment shown. b. Find the y-intercepts of each segment. c. Write the system of linear equations that will give you the coordinates

of the point of intersection of the two segments. d. Solve the system of equations.

LESSON 10: SOLVING PROBLEMS



EXERCISES

• Read through your Self Check and think about your work in this lesson.

• Write down what you have learned during the lesson.

• What would you do differently if you were starting the Self Check task now?

• Which method would you prefer to use if you were doing the task again? Why?

• Compare the new approaches you learned with your original method.

• Record your ideas — keep track of problem-solving strategies.

• Complete any exercises from this unit you have not finished.

LESSON 11: PUTTING IT TOGETHER

lesson 1: the race exercises

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