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Unit 3: Linear Equations

Notes

Date Topic/Assignment HW

Page

Due Date

3-A Linear Equations Introduction

3-B Intercepts of Linear Equations

3-C Slope of Lines

3-D Discovering Slope-Intercept Form

3-E Slope-Intercept Form

3-F Writing Linear Equations

3-G Point-Slope Form

3-H Point Slope Form Part 2

3-I Linear Inequalities

3-J Linear Equations with Restricted Domains

Review

Unit 3 Test

For additional resources, check out the course website at mathwithmills.weebly.com

Name:

Period:

2

Warm-Up Date:

Solve the following equations:

a. 2

5−

3𝑥

10=

3

4 b. Solve for y: 6𝑥 − 3𝑦 − 4 = 8𝑥 + 8

c. 6𝑥 + 2(𝑥 + 4) = 3(𝑥 − 6) + 11

Warm-Up Date:

Cammy has 8 more dollars than Liz. Liz has twice as much money as Tilley. How much money does

each person have if the sum of their money is $63?

Warm-Up Date:

. Solve each of the following inequalities for the given variable. Represent your solutions on a number

line.

a. 2(2𝑝 − 3) > −20 b. −15 ≤ 3𝑘 + 9 ≤ 18

3

Warm-Up Date:

GO WOLFPACK!

The graph below describes the distance two cars have traveled after

leaving a football game at the University of Nevada.

1. Which car was traveling faster? How can you tell?

2. The lines cross at (2, 80). What does this point represent?

3. Assuming that Car A continued to travel at a constant rate, how far did Car A travel in

the first 4 hours?

Warm-Up Date:

Evaluate the following expressions given the values below.

a. ab + bc + ac for a = 2, b = 5, and c = 3 b. for x = −2 and y = 6

4

Warm-Up Date:

Are the following sequences Arithmetic, Geometry, or neither?

a. 17, 12, 7, 2, … b. 7, 10, 15, 22, 31, … c. 3, 6, 12, 24, 48, …

Reed High School’s population is 250 students fewer than twice the population of Wooster High

School. The two schools have a total of 2858 students. How many students attend Reed High School? How about Wooster High School?

Warm-Up Date:

Use the Order of Operations to simplify the following expressions.

a. 5 − 2 · 32 b. (−2)2 c. 18 ÷ 3 · 6

d. −22 e. (5 − 3)(5 + 3) f. 24 ∙1

4÷ (−2)

5

Warm-Up Date:

Solve the following equations or inequalities:

a. Solve for y: 5𝑥 + 4𝑦 − 9 = 8𝑥 + 11 b. 3𝑥

6−

5

8=

11

12

b. −10 ≤ 3𝑥 + 5 < 26

Warm-Up Date:

Are the following sequences Arithmetic, Geometry, or neither?

a. 7, 12, 17, 22, … b. 24, 12, 6, 3, 1.5, … c. 1, 2, 3, 5, 8, …

3. Calculate the following values using the Order of Operations. Show your steps.

a. (−4)(−2) − 6(2 − 5) b. 23 − (17 − 3 · 4)2 + 6

c. 14(2 + 3 − 2 · 2) ÷ (42 − 32) d. 12.7 − 18.5 + 15 + 6.3 − 1 + 28.5

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Warm-Up Date:

Complete the diamond problems below:

Warm-Up Date:

1. Are the following sequences Arithmetic, Geometry, or neither?

a. -2, 1, 4, 7, … b. 9, 5, 1, -3, … c. 3, 9, 27, 81 …

2. Evaluate the expressions below for the given values.

a. 30 − 2x for x = −6 b. x2 + 2x for x = − 3

c. − x + 9 for x = −6 d. for k = 9

Warm-Up Date:

A board 39 inches long is cut into two pieces. One piece is three centimeters more than twice the

second piece. How long is each board?

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Warm-Up Date:

. Use the table to graph the rule below.

𝑦 = 4 − 𝑥2

1. What does your graph look like?

Warm-Up Date:

. Are the following sequences Arithmetic, Geometry, or neither?

a. 3, 5, 8, 12, … b. 4,5,9,14, … c. 5, 1, -3, -7, …

I am thinking of a number (again!). The sum of my number and 7 is the same as 5 less than three times

my number. What is it?

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3-A: Linear Equations Introduction

Other forms:

1. Example

a. b.

2. Example

a. b.

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3. Example

a. b.

4. Guided Practice

a. Decide if the ordered pair is a solution. b. Complete the ordered pair.

c. Complete the table.

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5. Example

A. B.

C. D. E.

6. Example

7. Guided Practice

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3-B: Intercepts of Linear Equations

Finding Intercepts Standard Form

Graphing by plotting the intercepts

1. Example 2. Guided Practice

3. Example

The equation does not have to be in standard form 4. Example

On the same coordinate grid, graph x-2 = 0 and -3y = -6

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Horizontal and Vertical Lines

5. Guided Practice

On the same coordinate grid, graph the following:

a. x + 3 = -1 b. 2x = 4

c. y – 5 = 0 d. -3y = 15

6. Example

-

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3-C: Slope of Lines

Slope

1. Example

Find the slope of each line.

2. Guided Practice.

Matching.

What about horizontal and vertical lines?

3. Example 4. Guided practice

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Find the slope. Find the slope.

a. b. a. b. (3, -2), (-5, -2)

c.

5. Example

Find the slope.

a. b. c. −4𝑥 − 3𝑦 = 7 d. x = -3

6. Guided Practice

Find the slope

a. 3𝑥 − 7𝑦 = 12 b. 4𝑥 + 8𝑦 = −13 c. −5𝑥 − 2𝑦 = 17 d. y = 2

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3-D Discovering Slope-Intercept Form

1. Find the slope of each line below.

a. b. c.

2. Graph each line with the given slope that goes through the given point.

a. m = 23; (-1, 2) b. m = – 2

1; (3, -4) c. m = – 3

5; (-4, -3)

3. Explain how you graphed the lines from problem 2.

4. The graphs of four lines are shown below. Complete the table for each graph.

Equation Slope y-intercept

y = -2x - 4

y = 2x + 3

y = 2x

y = 1

2 x – 2

a. Compare the equation of each line with its slope.

Describe the relationship(s) you see.

b. Compare the equation of each line with its y-intercept. Describe the relationship(s) you see.

x- 5 5

y

- 5

5

y = 2 x + 3 y = 2 x

y = 12

x – 2

y = -2 x – 4

x

y

x

y

x

y

x- 5 5

y

- 5

5

x- 5 5

y

- 5

5

x- 5 5

y

- 5

5

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5. a. Use an x-y table to graph the line y = 12 x – 4

Compare the equation of the line and the y-intercept.

Describe the relationship(s) you see.

b. Find the slope of the line you graphed. Compare the

equation of the line and the slope. Describe the

relationship(s) you see.

6. Look back at your answers to problems 4 and 5. Describe how you could use this information to identify the

slope and y-intercept of any line from its equation. Use the equation y = -3x + 2 as an example.

To find the slope of y = -3x + 2, I would …

To find the y-intercept of y = -3x + 2, I would …

7. Use your instructions in #6 to identify the slope and y-intercept and graph the following:

a. y = -3x + 2 b. y = 2

5 x – 3

slope = y-int. = slope = y-int. =

x- 5 5

y

- 5

5

x- 5 5

y

- 5

5

x- 5 5

y

- 5

5

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3-E: Slope-Intercept Form

1. Example

Identify the slope and y-intercept of each equation.

a. 𝑦 = 2𝑥 − 1 b. 𝑦 =−2

3𝑥 + 7 c. 𝑦 = −3𝑥 −

1

2 d. 𝑦 =

7

4𝑥 +

3

4

2. Guided Practice

Identify the slope and y-intercept of each equation.

a. 𝑦 =−1

2𝑥 −

5

3 b. 𝑦 = 5𝑥 +

1

2 c. 𝑦 = −3𝑥 − 8 d. 𝑦 = −

3

4𝑥 − 4

3. Example

Graph each line using the slope and the y-intercept.

a. 𝑦 = −3

4𝑥 − 4 b. 𝑦 = 2𝑥 + 1 c. 𝑦 = −

2

3𝑥

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4. Guided Practice

Graph each line using the slope and the y-intercept.

a. 𝑦 =3

5𝑥 − 2 b. 𝑦 = −3𝑥 + 1 c. 𝑦 = 2𝑥

5. Example 6. Guided Practice

Graph the horizontal/vertical lines.

a. 3x = -12 b. y + 1 = -3 a. On same grid graph: 2 + x = 0, 2y =4

7. Example 8. Guided Practice

Write the equation of the line in slope-intercept form.

a. b. a.

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3-F: Writing Linear Equations

1. Example

2. Guided Practice

3. Example 4. Guided Practice

Graph the line by using the slope and y-intercept. Graph the line by using the slope and y-intercept.

a. 𝑦 = 2𝑥 = 3 b. 2𝑥 + 3𝑦 = −6 a. 𝑦 =1

2𝑥 − 2 b. 3𝑥 − 4𝑦 = 12

5. Example

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6. Guided Practice

Graph each line described.

a. Through the point (-2, 3) with slope ½ b. Through the point (3, -1) with slope -2

7. Example

You can use the grid to help you.

8. Guided Practice

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3-G: Point-Slope Form

1. Example

a. b.

2. Guided Practice

3. Example

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4. Guided Practice

a. (5, -3), 𝑚 =2

5 b. (3, -1), 𝑚 = −

4

3 c.

5. Example

Write an equation for the line described.

a. horizontal through (2, -3) b. vertical through (2, -3) c. through (-4, -5), m = 0

d. through (3, 8), m = undefined

6. Guided Practice

Write an equation for the line described.

a. though (2, -5), m = 0 b. vertical though ( -3, 10)

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3-H: Point-Slope Form Part 2

1. Example

Write an equation of the line through the points (-2, 5) and (3, 4). Give the final answer in slope intercept form.

2. Guided Practice

Write an equation of the line through the points (4, 1) and (6, -2). Give the final answer in slope intercept form.

Summary of Equations of Lines

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3. Example

Write the equation for the line described.

a. slope is -3, y-intercept is 2 b. vertical, through (4, -6) c. through (3, -9), slope is 0

d. through (-4, -1) and (2, 2) e. through (3, -4), 𝑚 =2

3

4. Guided Practice

Write the equation for the line described.

a. slope is 4, y-intercept is -3 b. horizontal, through (-2, 5) c. through (-2, 8), slope is undefined

d. through (-1, 5) and (3, 1) e. through (5, -3), 𝑚 = −2

5

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3-I: Linear Inequalities in Two Variables

Inequality Symbols and Rules

1. Example

Graph each inequality.

a. 𝑦 <2

3𝑥 − 1 b. 𝑦 ≥ −2𝑥 + 1 c. 𝑦 ≤ −

2

3𝑥

2. Guided Practice.

Graph each inequality.

a. 𝑦 ≥ −1

3𝑥 + 1 b. 𝑦 <

5

3𝑥 c. 𝑦 > −

3

4𝑥 − 2

What if the equation is given in standard form? What about horizontal and vertical lines?

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3. Example

Graph each inequality.

a. 2𝑥 + 3𝑦 ≥ −6 b. 𝑦 < 3 c. 𝑥 ≥ −1

4. Guided Practice

Graph each inequality.

a. −2𝑥 + 5𝑦 ≥ 10 b. 𝑦 > −2 c. 𝑥 ≤ 4

5. Example 6. Guided Practice

Write the inequality for the graph shown.

a. b. a.

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3-J: Linear Equations with Restricted Domains

What is a Restricted Domain?

1. Example

Graph the following.

a. 𝑦 =1

2𝑥 − 1 𝑖𝑓 𝑥 ≥ −2 b. 𝑦 = −

2

3𝑥 − 3 𝑖𝑓 𝑥 < −3 c 𝑦 = 2 𝑖𝑓 𝑥 > 1

2. Guided Practice

Graph the following.

a. 𝑦 = 2𝑥 − 2 𝑖𝑓 𝑥 ≥ 1 b. 𝑦 = −𝑥 + 1 𝑖𝑓 𝑥 < 2 c 𝑦 = −3 𝑖𝑓 𝑥 ≤ 3

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3. Example

Graph the following.

a. 𝑦 = −2𝑥 + 1 𝑖𝑓 − 2 < 𝑥 ≤ 1 b 𝑦 = 𝑥 − 2 𝑖𝑓 0 ≤ 𝑥 < 4

4. Guided Practice

Graph the following.

a. 𝑦 = −2

3𝑥 − 4 𝑖𝑓 − 3 < 𝑥 ≤ 3 b 𝑦 = 𝑥 + 1 𝑖𝑓 − 1 ≤ 𝑥 < 3

5. Example 6. Guided Practice

Write the equation with domain for the graph shown.

a. b. c. a.