Chapter 3

Linear Equations

3-1: Graphing Linear Equations• Linear Equation:

• Standard Form:

• Constant:

• Coefficient:

• Label the parts of the equation:

5x + 7y = 4

Ex 1: Determine whether each equation is a linear equation. Write the equation in standard form.

a. y = 4 – 3x Guided Practice 1a.

b. 6x – xy = 4 Guided Practice 1b.

Page 153

• x-intercept:

• y-intercept:

Page 154Ex 2: Find the x and y intercepts on the line graphed at the right

Guided Practice 2a: Find the x and y intercepts on the line graphed at the right

Draining a Pool

Time (h) Volume (gal)

X Y

0 10,080

2 8,640

6 5,760

10 2,880

12 1,440

14 0

Page 155Ex 3:

a. Find the x and y-intercepts of the graph of the function

b. Describe what the intercepts mean in this situation

Draw graph here:

Page 155Guided Practice 3A: The table shows the function relating the distance to an amusement park in miles and the time in hours the Torrez family has driven. Find the x- and y-intercepts. Describe what they intercepts mean in this situation. Time (h) Distance (mi)

0 248

1 186

2 124

3 62

4 0

y = 2 x = 2

Domain:

Range:

(Page 155)Ex 4: Graph 2x + 4y = 16 by using the x- and y-intercepts.

(Page 155)Guided Practice 4A: -x + 2y = 3 Guided Practice 4B: y = -x -5

Page 156Ex 5: Graph

X 1/3x+2

Y (x, Y)

-3 1/3(-3) + 2

1 (-3, 1)

0 1/3(0) + 2

2 (0, 2)

3 1/3(3) + 2

3 (3, 3)

6 1/3(6) + 2

4 (6, 4)

Page 156Guided Practice: Graph each equation by making a table.

5A. 2x – y = 2 5B. x = 3

HW: 157-159, 13-49 (odd), 42, 50, 51-57 (odd), 58

3-2: Solving Linear Equations by Graphing Page 161

• Linear Function:

• Root

• Zeros

Page 162Ex 1: Solve each equation

a. 0 = 1/3x – 2 b. 3x + 1 = -2 (Solve by graphing – use table)

Guided Practice 1A. 0 = 2/5x + 6 1B. -1.25x + 3 = 0 (Solve by graphing)

Page 162-163Ex 2: Solve each equation

a. 3x +7 = 3x + 1 b. 2x – 4 = 2x – 6 (Solve by graphing)

Guided Practice:2A. 4x + 3 = 4x – 5 2B. 2 – 3x = 6 – 3x (Solve by

graphing)

Page 163Ex 3: Estimating by Graphing

Emily is going to a local carnival. The function m = 20 – 0.75r represents the amount of money, m, she has left after r, rides. Find the zero of this function. Describe what this value means in this context.

R M = 20 – 0.75r m (r,m)

Page 163Guided Practice 3

Antoine’s class is selling candy to raise money for a class trip. They paid $45 for the candy, and they are selling each candy bar for $1.50. The function y = 1.50x – 45 represents their profit, y, when they sell x candy bars. Find the zero and describe what it means in the context of this situation.

X Y = 1.50x - 45 y (x,y)

HW: pg.164-165: 11-43 (odd), 36, 44-46, 48-49

Rise Run RiseRun

1. Original set up:

2. Move books to make ramp steeper:

3. Add more books:

4. #41 from page 177:

5. #42 from page 177:

3-3: Rate of Change and SlopePage 170

Rate of Change:

Change in yChange in x

X:

Y:

Positive rate of change:

Negative rate of change:

Ex 1: Use the table to find the rate of change. Then explain its meaning.

Guided Practice: The table shows how the tiled surface area changes with the number of floor tiles.

A. Find the rate of changeB. Explain the meaning of the rate of change

Number of Floor Tiles

Area of Tiled surface

X Y

3 48

6 96

9 144

Number of

Computer Games

Total Cost ($)

X Y

2 78

4 156

6 234

Page 171Ex 2: The graph shows the number of people who visited U.S. them parks in recent years.

a. Find the rates of change for 2000-20002 and 2002-2004\

b. Explain the meaning of the rate of change in each case.

c. How are the different rates of change shown on the graph?

Guided Practice: Refer to the graph above. Without calculating, find the 2-year period that has the least rate of change. Then calculate to verify your answer.

Page 172Ex 3: Determine whether each function is linear. Explain.

Guided Practice 3A: Guided Practice 3B:

X Y

1 -6

4 -8

7 -10

10 -12

13 -14

X Y

-3 10

-1 12

1 16

3 18

5 22

X Y

-3 11

-2 15

-1 19

1 23

2 27

X Y

12 -4

9 1

6 6

3 11

0 16

Slope (m) =

• Slope is a type of _______________________

• Slope describes _________________________________

• The greater the ______________________________ of the slope, the steeper the line.

• m = change in y or ___________ change in x

Slope can be:

_________________ _________________

_________________ _________________

Ex 5: Find the slope of the line that passes through ( -2, 4 ) ad ( -2, -3 ).

Guided practice 5A:

Find the slope of the line that passes through ( 6, 3 ) and ( 6, 7 ).

HW: 175-177, 15-39 (odd), 42-49, 52

Page 174Ex 6: Find the value of r so that the line through ( 1, 4 ) and ( -5, r ) has a slope of 1/3

Guided Practice: Find the value of r so the line that passes through each pair of points has the given slope.

6A. ( -2, 6 ), ( r, -4 ); m = -5 GP 6B: ( r, -6 ), (5, -8 ); m = -8

3-4: Direct VariationPage 180

Direct variation:

Constant of variation/Constant of proportionality =

Ex 1: Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points

a. b.

Page 180

Guided Practice1a: Name the constant of variation for y = 1/4x. Then fid the slope of the line that passes through ( 0, 0) ad ( 4, 1 ).

1B: Name the constant of variation for y = -2x. Then find the slope of the line that passes through ( 0,0) ad ( 1, -2 ).

Page 181Ex 2: Graph y = -6x

***The graph of a direct variation will always pass through the origin (0,0)

Guided Practice:2A. y = 6x 2B. y = 2/3x 2D. y = -3/4x

If the relationship between the values of y and x can be described by a direct variation equation, then we say that y varies directly as x.

Ex 3: Suppose y varies directly as x, ad y = 72 when x = 8

a. Write a direct variation equation that relates x and y

b. Use the direct variation equation to find x when y = 63

Guided Practice 3: Suppose y varies directly as x, and y = 98 when x = 14. Write a direct variation equation that relates x and y. Then find y when x = -4

y = kxd = rt

Ex 4: The distance a jet travels varies directly as the umber of hours it flies. A jet traveled 3420 miles in 6 hours.a. Write a direct variation equation for the distance d flown in time t.

b. Graph the equation.

c. Estimate how many hours it will take for an airliner to fly 6500 miles

Guided Practice 4: A hot-air balloon's height varies directly as the balloon’s ascent time in minutes.

A. Write a direct variation for the distance d ascended in time t.

B. Graph the equation

C. Estimate how many minutes it would take to ascend 2100 feetD. About how many minutes would it take to ascend 3500 feet?

HW: Page 183-184, 10-33

3-5: Arithmetic Sequences as Linear Functions

Sequence:

Terms of the sequence:

Distance (m) 400 800 1200 1600 2000

Time (min:sec) 1:32 3:04 4:36 6:08 7:40

Arithmetic Sequence:

* Common difference (d):

Increasing:

Decreasing:

… means:

Ex 1: Determine whether each sequence is an arithmetic sequence. Explain.a. -4, -2, 0, 2 b. ½, 5/8, ¾, 13/16

Guided Practice1A. -26, -22, -18, -14 1B. 1, 4, 9, 25

Ex 2: Find the next three terms of the arithmetic sequence 15, 9, 3, -3

Guided Practice: Find the next four terms of the arithmetic sequence 9.5, 11, 12.5, 14

a:

d:

n:

y:

:

Ex 3: a. Write a equation for the nth term of the arithmetic sequence -12, -8, -4, 0, …

b. Find the 9th term of the sequence.

c. Graph the first five terms of the sequence

d. Which term of the sequence is 32?

Guided Practice: Consider the arithmetic sequence 3, -10, -23, -36, …

3A: Write an equation for the nth term of the sequence

3B: Find the 15th term in the sequence

3C. Graph the first five terms of the sequence

3D. Which term of the sequence is -114

Ex 4: Marisol is mailing invitations to her quinceanera. The arithmetic sequence, $0.42, $0.84, $1.26, $1.68, … represents the cost of postage.

a. Write a function to represent this sequenceb. B. Graph the function

Guided Practice: The chart below shows the length of Marin’s long jumps.

A. Write a function to represent this arithmetic sequence.

B. Then graph the function

Jump 1 2 3 4

Length (ft) 8 9.5 11 12.5

3-6: Proportional and Nonproportional Relationships

An equation is proportional when _________________________________________________

Ex 1: Marcos is a personal trainer at a gym. In addition to his salary, he receives a bonus for each client he sees.

a. Graph the data. What can you deduce from the pattern about the relationship between the number of clients and the bonus pay?

b. Write an equation to describe this relationship

c. Use this equation to predict the amount of Marcos’ bonus if he sees 8 clients

Guided Practice: A professional soccer team is donating money to a local charity for each goal they score.

A. Graph the data. What ca you deduce from the pattern about the pattern about the relationship between the number of goals and the money donated?

B. Write an equation to describe this relationship.C. use this equation to predict how much money will be donated for 12 goals

Number of Clients 1 2 3 4 5Bonus Pay ($) 45 90 135 180 225

Nonproportional Relationship:

Ex 2: Write an equation in function notation for the graph

Guided Practice: 2A. Write an equation in function notation for the relation shown in the table

2B. Write an equation in function notation for the graph

HW: Page 198-199 5, 7, 9-15, 17-35

X 1 2 3 4

Y 3 2 1 0