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Collection by: Rupinder Sekhon. E-mail the author

# Linear Equations: Homework

Module by: UniqU, LLC. E-mail the author

Summary: This chapter covers principles of linear equations. After completing this chapter students should be able to: graph a linear equation; find the slope of a line; determine an equation of a line; solve linear systems; and complete application problems using linear equations.

## GRAPHING A LINEAR EQUATION

Work the following problems.

### Exercise 1

Is the point (2, 3) on the line 5x2y=45x2y=4 size 12{5x - 2y=4} {}?

Yes

### Exercise 2

Is the point (1, – 2) on the line 6xy=46xy=4 size 12{6x - y=4} {}?

### Exercise 3

For the line 3xy=123xy=12 size 12{3x - y=12} {}, complete the following ordered pairs.

(2, ) ( , 6)

(0, ) ( , 0)

(2, –6) (6, 6)

(0,–12) (4, 0)

### Exercise 4

For the line 4x+3y=244x+3y=24 size 12{4x+3y="24"} {}, complete the following ordered pairs.

(3, ) ( , 4)

(0, ) ( , 0)

### Exercise 5

Graph y=2x+3y=2x+3 size 12{y=2x+3} {}

### Exercise 6

Graph y=3x+5y=3x+5 size 12{y= - 3x+5} {}

### Exercise 7

Graph y=4x3y=4x3 size 12{y=4x - 3} {}

### Exercise 8

Graph x2y=8x2y=8 size 12{x - 2y=8} {}

### Exercise 9

Graph 2x+y=42x+y=4 size 12{2x+y=4} {}

### Exercise 10

Graph 2x3y=62x3y=6 size 12{2x - 3y=6} {}

### Exercise 11

Graph 2x+4=02x+4=0 size 12{2x+4=0} {}

### Exercise 12

Graph 2y6=02y6=0 size 12{2y - 6=0} {}

### Exercise 13

Graph the following three equations on the same set of coordinate axes.

y=x+1   y=2x+1  y=x+1y=x+1   size 12{y=x+1}      {}y=2x+1   size 12{y=2x+1}      {}y=x+1 size 12{y= - x+1} {}
(1)

### Exercise 14

Graph the following three equations on the same set of coordinate axes.

y=2x+1  y=2x  y=2x1y=2x+1   size 12{y=2x+1} {}y=2x   size 12{y=2x} {}y=2x1 size 12{y=2x - 1} {}
(2)

### Exercise 15

Graph the line using the parametric equations

x=1+2t, y=3+tx=1+2t size 12{x=1+2t} {},y=3+t size 12{y=3+t} {}
(3)

### Exercise 16

Graph the line using the parametric equations

x=23t,   y=1+2tx=23t size 12{x=2 - 3t} {},  y=1+2t size 12{y=1+2t} {}
(4)

## SLOPE OF A LINE

Find the slope of the line passing through the following pair of points.

### Exercise 17

(2, 3) and (5, 9)

#### Solution

m=2m=2 size 12{m=2} {}
(5)

### Exercise 18

(4, 1) and (2, 5)

### Exercise 19

(– 1, 1) and (1, 3)

#### Solution

m=1m=1 size 12{m=1} {}
(6)

### Exercise 20

(4, 3) and (– 1, 3)

### Exercise 21

(6, – 5) and (4, – 1)

#### Solution

m=2m=2 size 12{m= - 2} {}
(7)

### Exercise 22

(5, 3) and (– 1, – 4)

### Exercise 23

(3, 4) and (3, 7)

#### Solution

m=undefinedm=undefined size 12{m="undefined"} {}
(8)

### Exercise 24

(– 2, 4) and (– 3, – 2)

### Exercise 25

(– 3, – 5) and (– 1, – 7)

#### Solution

m=1m=1 size 12{m= - 1} {}
(9)

### Exercise 26

(0, 4) and (3, 0)

Determine the slope of the line from the given equation of the line.

### Exercise 27

y=2x+1y=2x+1 size 12{y= - 2x+1} {}
(10)

#### Solution

m=2m=2 size 12{m= - 2} {}
(11)

### Exercise 28

y=3x2y=3x2 size 12{y=3x - 2} {}

### Exercise 29

2xy=62xy=6 size 12{2x - y=6} {}
(12)

#### Solution

m=2m=2 size 12{m=2} {}
(13)

### Exercise 30

x+3y=6x+3y=6 size 12{x+3y=6} {}
(14)

### Exercise 31

3x4y=123x4y=12 size 12{3x - 4y="12"} {}
(15)

#### Solution

m=3/4m=3/4 size 12{m=3/4} {}
(16)

### Exercise 32

What is the slope of the x-axis? How about the y-axis?

Graph the line that passes through the given point and has the given slope.

### Exercise 33

(1, 2) and m=3/4m=3/4 size 12{m= - 3/4} {}

### Exercise 34

(2, –1) and m=2/3m=2/3 size 12{m=2/3} {}

### Exercise 35

(0, 2) and m=2m=2 size 12{m= - 2} {}

### Exercise 36

(2, 3) and m=0m=0 size 12{m=0} {}

## DETERMINING THE EQUATION OF A LINE

Write an equation of the line satisfying the following conditions. Write the equation in the form y=mx+by=mx+b size 12{y= ital "mx"+b} {}.

### Exercise 37

It passes through the point (3, 10) and has slope=2slope=2 size 12{"slope"=2} {}.

#### Solution

y=2x+4y=2x+4 size 12{y=2x+4} {}
(17)

### Exercise 38

It passes through the point (4,5) and has m=0m=0 size 12{m=0} {}.

### Exercise 39

It passes through (3, 5) and (2, – 1).

#### Solution

y=6x13y=6x13 size 12{y=6x - "13"} {}
(18)

### Exercise 40

It has slope 3, and its y-intercept equals 2.

### Exercise 41

It passes through (5, – 2) and m=2/5m=2/5 size 12{m=2/5} {}.

#### Solution

y=2/5x4y=2/5x4 size 12{y=2/5x - 4} {}
(19)

### Exercise 42

It passes through (– 5, – 3) and (10, 0).

### Exercise 43

It passes through (4, – 4) and (5, 3).

#### Solution

y=7x32y=7x32 size 12{y=7x - "32"} {}
(20)

### Exercise 44

It passes through (7, – 2) and its y-intercept is 5.

### Exercise 45

It passes through (2, – 5) and its x-intercept is 4.

#### Solution

y=5/2x10y=5/2x10 size 12{y=5/2x - "10"} {}
(21)

### Exercise 46

Its a horizontal line through the point (2, – 1).

### Exercise 47

It passes through (5, – 4) and (1, – 4).

#### Solution

y=4y=4 size 12{y= - 4} {}
(22)

### Exercise 48

It is a vertical line through the point (3, – 2).

### Exercise 49

It passes through (3, – 4) and (3, 4).

#### Solution

x=3x=3 size 12{x=3} {}
(23)

### Exercise 50

It has x-intercept=3x-intercept=3 size 12{x"-intercept"=3} {} and y-intercept=4y-intercept=4 size 12{y"-intercept"=4} {}.

Write an equation of the line satisfying the following conditions. Write the equation in the form Ax+By=CAx+By=C size 12{ ital "Ax"+ ital "By"=C} {}.

### Exercise 51

It passes through (3, – 1) and m=2m=2 size 12{m=2} {}.

#### Solution

2xy=72xy=7 size 12{2x - y=7} {}
(24)

### Exercise 52

It passes through (– 2, 1) and m=3/2m=3/2 size 12{m= - 3/2} {}.

### Exercise 53

It passes through (– 4, – 2) and m=3/4m=3/4 size 12{m=3/4} {}.

#### Solution

3x4y=43x4y=4 size 12{3x - 4y= - 4} {}
(25)

### Exercise 54

Its x-intercept equals 3, and m=5/3m=5/3 size 12{m= - 5/3} {}.

### Exercise 55

It passes through (2, – 3) and (5, 1).

#### Solution

4x3y=174x3y=17 size 12{4x - 3y="17"} {}
(26)

### Exercise 56

It passes through (1, – 3) and (– 5, 5).

## APPLICATIONS

In the following application problems, assume a linear relationship holds.

### Exercise 57

The variable cost to manufacture a product is $25, and the fixed costs are$1200. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

#### Solution

y=25x+1200y=25x+1200 size 12{y="25"x+"1200"} {}
(27)

### Exercise 58

It costs $90 to rent a car driven 100 miles and$140 for one driven 200 miles. If xx size 12{x} {} is the number of miles driven and yy size 12{y} {} the total cost of the rental, write the cost function.

### Exercise 59

The variable cost to manufacture an item is $20, and it costs a total of$750 to produce 20 items. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

#### Solution

y=20x+350y=20x+350 size 12{y="20"x+"350"} {}
(28)

### Exercise 60

To manufacture 30 items, it costs $2700, and to manufacture 50 items, it costs$3200. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

### Exercise 61

To manufacture 100 items, it costs $32,000, and to manufacture 200 items, it costs$40,000. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

#### Solution

y=80x+24000y=80x+24000 size 12{y="80"x+"24000"} {}
(29)

### Exercise 62

It costs $1900 to manufacture 60 items, and the fixed costs are$700. If xx size 12{x} {} represents the number of items manufactured and yy size 12{y} {} the cost, write the cost function.

### Exercise 63

A person who weighs 150 pounds has 60 pounds of muscles, and a person that weighs 180 pounds has 72 pounds of muscles. If xx size 12{x} {} represents the body weight and yy size 12{y} {} the muscle weight, write an equation describing their relationship. Use this relationship to determine the muscle weight of a person that weighs 170 pounds.

#### Solution

y=2/5x;68y=2/5x size 12{y=2/5x} {};68

### Exercise 64

A spring on a door stretches 6 inches if a force of 30 pounds is applied, and it stretches 10 inches if a force of 50 pounds is applied. If xx size 12{x} {} represents the number of inches stretched, and yy size 12{y} {} the force applied, write an equation describing the relationship. Use this relationship to determine the amount of force required to stretch the spring 12 inches.

### Exercise 65

A male college student who is 64 inches tall weighs 110 pounds, and another student who is 74 inches tall weighs 180 pounds. Assuming the relationship between male students' heights xx size 12{ left (x right )} {}, and weights yy size 12{ left (y right )} {} is linear, write a function to express weights in terms of heights, and use this function to predict the weight of a student who is 68 inches tall.

#### Solution

y=7x338y=7x338 size 12{y = 7x - "338"} {} ; 138

### Exercise 66

EZ Clean company has determined that if it spends $30,000 on advertisement, it can hope to sell 12,000 of its Minivacs a year, but if it spends$50,000, it can sell 16,000. Write an equation that gives a relationship between the number of dollars spent on advertisement xx size 12{ left (x right )} {} and the number of minivacs sold yy size 12{ left (y right )} {}.

### Exercise 67

The freezing temperatures for Celsius and Fahrenheit scales are 0 degree and 32 degrees, respectively. The boiling temperatures for Celsius and Fahrenheit are 100 degrees and 212 degrees, respectively. Let CC size 12{C} {} denote the temperature in Celsius and FF size 12{F} {} in Fahrenheit. Write the conversion function from Celsius to Fahrenheit, and use this function to convert 25 degrees Celsius into an equivalent Fahrenheit measure.

#### Solution

F=9/5C+32F=9/5C+32 size 12{F=9/5C+"32"} {}; 77ºF

### Exercise 68

By reversing the coordinates in the previous problem, find a conversion function that converts Fahrenheit into Celsius, and use this conversion function to convert 72 degrees Fahrenheit into an equivalent Celsius measure.

### Exercise 69

The population of California in the year 1960 was 17 million, and in 1995 it was 32 million. Write the population function, and use this function to find the population of California in the year 2010. (Hint: Use the year 1960 as the base year, that is, assume 1960 as the year zero. This will make 1995, and 2010 as the years 35, and 50, respectively.)

#### Solution

y=3/7x+17;38.43y=3/7x+17 size 12{y=3/7x+"17"} {};38.43

### Exercise 70

In the U. S. the number of people infected with the HIV virus in 1985 was 1,000, and in 1995 that number became 350,000. If the increase in the number is linear, write an equation that will give the number of people infected in any year. If this trend continues, what will the number be in 2010? (Hint: See previous problem.)

### Exercise 71

In 1975, an average house in San Jose cost $45,000 and the same house in 1995 costs$195,000. Write an equation that will give the price of a house in any year, and use this equation to predict the price of a similar house in the year 2010.

### Exercise 81

A break-even point is the intersection of the cost function and the revenue function, that is, where the total cost equals revenue. Mrs. Jones Cookies Store's revenue and cost in dollars for xx size 12{x} {} number of cookies is given by R=.80xR=.80x size 12{R= "." "80"x} {} and C=.05x+3000C=.05x+3000 size 12{C= "." "05"x+"3000"} {}. Find the number of cookies that must be sold so that the revenue and cost are the same.

#### Solution

x=4000x=4000 size 12{x="4000"} {}
(32)

### Exercise 82

A company's revenue and cost in dollars are given by R=225xR=225x size 12{R="225"x} {} and C=75x+6000C=75x+6000 size 12{C="75"x+"6000"} {}, where xx size 12{x} {} is the number of items. Find the number of items that must be produced to break-even.

### Exercise 83

A firm producing computer diskettes has a fixed costs of $10,725, and variable cost of 20 cents a diskette. Find the break-even point if the diskettes sell for$1.50 each.

(8250, 12375)

(51)

### Exercise 112

A car rental company offers two plans. Plan I charges $12 a day and 12 cents a mile, while Plan II charges$30 a day but no charge for miles. If you were to drive 300 miles in a day, which plan is better? For what mileage are both rates the same?

150 miles

### Exercise 113

The supply curve for a product is y=250x1,000y=250x1,000 size 12{y="250"x - 1,"000"} {} and the demand curve for the same product is y=350x+8,000y=350x+8,000 size 12{y= - "350"x+8,"000"} {}, where xx size 12{x} {} is the price and yy size 12{y} {} the number of items produced. Find the following.

12,500

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