# Graphs and Linear Equations

Section 3.1

Use the given equation to find the missing coordinates in the following tables:

34.

X Y

-2

0

2

-3

-7

Graph each equation. Plot at least five points for each equation. Use graph paper. See Examples 3.5. If you have a graphing calculator, use it to check your graphs when possible.

58.

60.

94. Demand Equation. Helen's Health Food usually sells 400 cans of ProPac Muscle Punch per week when the price is $5 per an. After experimenting with prices for some time, Helen has determined that the weekly demand can be found by using the equation , where d is the number of cans and p is the price per can.

a) Will Helen sell more or less Muscle punch if she raises her prices from $5?

b) What happens to her sales every time she raises her price $1?

c) Graph the equation.

d) What is the maximum price that she can charge and still sell at least one can?

Section 3.2

Find the slope of each line

10.

16.

Solve each problem:

40.

The line through (-2, 5) with slope -1

50.

Draw through (-4, 0) and (0, 6). What is the slope of any line parallel to ? Draw through the origin and parallel to .

62. Retirement Pay. The annual Social Security benefit of a retiree depends on the age at the time of retirement. The accompanying graph gives the annual benefit for persons retiring at ages 62 through 70 in the year 2005 or later (Social Security Administration, www.ssa.gov).

What is the annual benefit for a person who retires at age 64? At what retirement age does a person receive an annual benefit of $11,600?

Fine the slope of each line segment on the graph, and interpret your results.

Why do people who postpone retirement until 70 years of age get the highest benefit?

Section 3.3

Write an equation for each line. Use slope-intercept form if possible.

10.

Find the slope and y-intercept for each line that has a slope and y-intercept.

30.

X + 2y = 3

Write each equation in standard form using only integers.

46.

Draw the graph of each line using its y-intercept and its slope.

62.

In each case determine whether the lines are parallel, perpendicular, or neither.

74.

Solve each problem

90. Marginal Revenue. A defense attorney charges her client $4000 plus $120 per hour. The formula R = 120n + 4000 gives her revenue in dollars for n hours of work.

What is her revenue for 100 hours of work?

What is her revenue for 101 hours of work?

By how much did the one extra hour of work increase the revenue? (The increase in revenue is called the marginal revenue for the 101st hour.)

92. Single Women. The percentage of women in the 20 - 24 age group who have never married went from 55% in 1970 to 73% in 2000 (Census Bureau, www.census.gov). Let 1970 be year 0 and 2000 be year 30.

a. Find and interpret the slope of the line through the points (0, 55) and (30, 73).

b. Find the equation of the line in part (a).

c. What is the meaning of the y-intercept?

d. Use the equation to predict the percentage in 2010.

e. If this trend continues, then in what year will the percentage of women in the 20 - 24 age group who have never married reach 100%?

94. Pens and Pencils. A bookstore manager plans to spend $60 on pens at 30 cents each and pencils at 10 cents each. The equation 0.10x + 0.30y = 60 can be used to model the situation.

a. What do x and y represent?

b. Graph the equation.

c. Write the equation in slope-intercept form.

d. What is the slope of the line?

e. What does the slope tell you?

Section 3.4

Write each equation in slope-intercept form.

10.

Find the equation of the line that goes through the given point and has the given slope. Write the answer in slope-intercept form.

22.

46. The lines in each figure are perpendicular. Find the equation (in slope-intercept form) for the solid line.

Find the equation of each line. Write each answer in slope-intercept form.

54. The line is parallel to -3x + 2y = 9 and contains the point (-2, 1).

Find the equation of each line in the form y = mx + b if possible.

64. The line through the origin that is perpendicular to the line through (-3, 0) and (0, -3).

Solve each problem.

78. Direct Deposit. The percentage of workers receiving direct deposit of their paychecks went from 32% in 1994 to 60% in 2004 (www.directdeposit.com) . Let 1994 be year 0 and 2004 be year 10.

Write the equation of the line through (0, 32) and (10, 60) to model the growth of direct deposit.

Use the graph on the next page to predict the year in which 100% of all workers will receive direct deposit of their paychecks.

Use the equation from part (a) to predict the year which 100% of all workers will receive direct deposit.

80. Age at first marriage. The median age at first marriage for females increased from 24.5 hears in 1995 to 25.1 years in 2000 (U.S. Census Bureau, www.census.gov). Let 1995 be year 5 and 2000 be year 10.

Find the equation of the line through (5, 24.5) and (10, 25.1).

What do x and y represent in your equation?

Interpret the slope of this line.

In what year will the median age be 30?

Graph the equation.

92. Basal energy requirement. The basal energy requirement B is the number of calories that a person needs to maintain the life process. For a 28-year-old female with a height of 160 centimeters and a weight of 45 kilograms (kg), B is 1300 calories. If her weight increases to 50 kg, then B is 1365 calories. There is a linear equation that expresses B in terms of her weight w. Find the equation and find the basal energy requirement if her weight is 53.2 kg.

Section 3.6

Graph each inequality

26.

36.

46.

Solve each problem:

50. Maple Rockers. Ozark Furniture Company can obtain at most 3000 board feet of maple lumber for making its classic and modern maple rocking chairs. A classic maple rocker requires 15 board feet of maple, and a modern rocker requires 12 board feet of maple. Write an inequality that limits the possible number of maple rockers of each type that can be made, and graph the inequality in the first quadrant.

52. Enzyme concentration. A food chemist tests enzymes for their ability to break down pectin in fruit juices (Dennis Callas, Snapshots of Applications in Mathematics). Excess pectin makes juice cloudy. In one test, the chemist measures the concentration of the enzyme, c, in milligrams per milliliter and the fraction of light absorbed by the liquid, a. If a > 0.07c + 0.02, then the enzyme is working as it should. Graph the inequality in the first quadrant.

2.1 Solve each equation

8.

2.2 Solve each equation

18.

2.6 Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity.

26.

42. 0.6

50.

Build-up each fraction so it is equal to the original fraction:

1.

2.

Reduce each fraction to it's lowest terms:

3.

4.

Evaluate the following expressions:

5. 9 - ( -2) + ( 3 - 4 )

6.

7.

8.

Simplify the following expressions:

9. 2(3a - 5b) - 2(6a - b)

10. 8x - 4[5 - 3(x+1)]

11.

Burke and Nora deliver pizzas for Racine's Pizza. Burke averages one delivery every 0.25 hours and Nora averages one delivery every 0.20 hours. At what rate (in deliveries per hour) are the deliveries m

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