Section 2.3 Solving Linear Equations

TRUE/FALSE

1. Linear equations are always difficult to solve.

ANS: F PTS: 1 DIF: easy

2. To say that an equation is linear means that it is a straight line.

ANS: F PTS: 1 DIF: easy

3. Logarithms are sometimes needed to solve linear equations.

ANS: F PTS: 1 DIF: easy

4. Techniques for solving linear equations can sometimes be used to reverse the roles of variables in equations.

ANS: T PTS: 1 DIF: easy

Multiple Choice

1. An iPad costs $491 at your local computer store. The cable company charges $24 per month for a DSL Internet connection. Write a formula that gives the cost , in dollars, of buying an iPad and paying for months of Internet service.

a.

c.

b. d.

ANS: A PTS: 1 DIF: easy

2. The life expectancy , in years, for a child born years after 2000 is given by a. 2005 b. 2004 c. 2002 d. That life expectancy never occurs.

In what birth year would a child have a life expectancy of 77.34 years?

ANS: B PTS: 1 DIF: easy a. c. b. d.

3. A rental car charges $72.87 for insurance and other fees plus a flat rate of $24.95 per day. Write a formula that gives the cost , in dollars, of renting a car for days.

ANS: A PTS: 1 DIF: easy a. 7.38 weeks b. 52.71 weeks

4. One rental company charges $52.71 in fees for a television plus $22.77 per week. A second rental company charges $31.75 in fees plus $25.61 per week. How many weeks would make the total cost of renting a television the same for both companies? You can be charged for part of a week, so you should report your answer correct to two decimal places.

ANS: A PTS: 1 c. 10.28 weeks d. DIF: It never happens medium a. 2015 b. 2012

5. The amount , in billions of dollars, collected by the Internal Revenue Service years after 2003 is given by According to this model, in what year would IRS collections reach 3554 billion dollars?

ANS: B PTS: 1 c. 2010 d. DIF: It never happens medium a. 26 songs b. 22 songs c. 23 songs d. 25 songs

6. You got a birthday gift of $45.64, which you choose to spend on music.You pay $12.75 for an iPad app that lets you download songs for $1.43 each. How many songs can you download before your birthday money is used up?

ANS: C PTS: 1 DIF: medium a. In the 13th Olympics after 1900 b. In the 12th Olympics after 1900 c. In the 16th Olympics after 1900 d. The record never reached 9.02 meters.

7. The long jump record , in meters, for the th Olympics after 1900 is given approximately by the formula . When did the long jump record reach 9.02 meters?

ANS: A PTS: 1 DIF: medium a. 13.16 pounds b. 22.39 pounds

8. If you wish to have an item you ordered delivered quickly, you must pay extra for shipping, and the amount you pay depends on the weight , in pounds, of the item. The shipping cost , in dollars, is given by .

You have $33.78 to spend on shipping costs. How large an item can you have shipped?

ANS: D PTS: 1 c. 13 pounds d. DIF: 11.39 pounds medium

9. The cost C, in dollars, of downloading tunes from an Internet music service is given by a. b. c. d. There is no solution.

Here a and b are constants. Solve this equation for in order to express the number of tunes you can download as a function of the cost C.

ANS: B PTS: 1 DIF: medium a. 6.14 kilometers b. 8.04 kilometers

10. The speed , in meters per second, of sound in the ocean depends on the depth , in kilometers. (It depends on other quantities as well, but those will be ignored for this problem.) The relationship is .

At what depth is the speed of sound 1633.18 meters per second?

ANS: D PTS: 1 c. 29461.38 kilometers d. DIF: 5.8 kilometers medium a. 5.37 months b. 17.14 months c. 6.79 months d. A baby whale is never that long.

11. The length , in feet, of a baby blue whale depends on its age , in months. The relationship is .

How old is a baby wale that is 44 feet long?

ANS: A PTS: 1 DIF: medium

12. You start making regular deposits to your piggy bank. The amount , in dollars, in your piggy bank after weeks is given by

The price of the computer you want to buy is decreasing. The price , in dollars, weeks after you started saving money is given by a. After 8 weeks b. After 7 weeks

When will there be enough money in your piggy bank to buy the computer? Report your answer as a whole number.

ANS: A PTS: 1 c. d. After 10 weeks There will never be enough.

DIF: medium

13. For connection to the system and water usage , in hundreds of gallons, water company A charges a. The charge will never be the same. b. 394 hundred gallons c. 190.43 hundred gallons d. 394.12 hundred gallons

For connection to the system and water usage , in hundreds of gallons, water company B charges .

Both and are in dollars. How many hundreds of gallons of water will result in the same charge from water company A and water company B?

ANS: D PTS: 1 DIF: medium a. There is no solution. b. c. d.

14. For connection to the system and water usage , in hundreds of gallons, a water company charges dollars.

Solve this equation for to express water consumption as a function of the charges.

ANS: C PTS: 1 DIF: medium a. b. c. d.

15. As you drive up a hill, your elevation , in feet, increases with each mile that you drive. The relationship is . Express the miles you drive as a function of elevation.

ANS: A PTS: 1 DIF: medium a. There is no solution. b. c. d.

16. Solve the equation .

ANS: D PTS: 1 DIF: medium a. There is no solution. b. c. d.

17. Solve the equation .

ANS: D PTS: 1 DIF: medium

18. Solve the equation a. There is no solution. c. b. d.

ANS: B PTS: 1 DIF: medium

19. Solve the equation for a. b. d.

ANS: C PTS: 1 DIF: medium a. There is no solution. b. c. d.

20. Solve the equation .

ANS: D PTS: 1 DIF: medium a. There is no solution. b. c. d.

21. Solve the equation for .

ANS: B PTS: 1 DIF: medium

22. Solve the equation for a. There is no solution. b. c. d.

ANS: B PTS: 1 DIF: medium a. There is no solution. b. c. d.

23. Solve the equation for y.

ANS: D PTS: 1 DIF: medium a. There is no solution. c. c.

24. Solve the equation for .

ANS: D PTS: 1 DIF: medium

25. Solve the equation for a. There is no solution. b. c. d.

ANS: B PTS: 1 DIF: medium

26. Solve the equation for a. There is no solution. c. b. d.

ANS: B PTS: 1 DIF: medium a. There is no solution. c. b. d.

27. Solve the equation for .

ANS: B PTS: 1 DIF: medium

28. Solve the equation for a. There is no solution. c. b. d.

ANS: C PTS: 1 DIF: medium a. There is no solution. c. b. d.

29. Solve the equation for z.

ANS: C PTS: 1 DIF: medium

30. Solve the equation for z a. There is no solution. c. b. d.

ANS: C PTS: 1 DIF: medium a. There is no solution. c. b. d.

31. The value V, in dollars, of an automobile years after 2010 is given by the formula .

Solve this equation for to express the time in years since 2010 as a function of the value of the car.

ANS: C PTS: 1 DIF: medium

32. Solve the following equation for a. There is no solution. c. b. d.

ANS: D PTS: 1 DIF: medium

Essay

1. Music source A charges a monthly fee of $6.17 plus 99 cents per song. Music source B charges a monthly fee of $10.17 plus 79 cents per song.

A: Find a formula that gives the monthly cost , in dollars, of buying songs from Company A as a function of the number of songs you buy in a month.

B: Find a formula that gives the monthly cost , in dollars, of buying songs from Company B as a function of the number of songs you buy in a month.

C: What number of songs purchased will give the same monthly cost for Company A and Company B?

ANS: A:

B:

C: The cost is the same if you buy 20 songs per month.

PTS: 3 DIF: medium

2. An electric company charges a connection fee of $54 plus $0.19 for each kilowatt-hour of electricity used.

A: Write a formula that gives the cost , in dollars, of connecting and using kilowatt-hours of electricity.

B: Solve the equation you found in part A to express the kilowatt-hours used as a function of the cost E.

C: How many kilowatt-hours are used if the total charge (connection plus charge for electricity usage) is $152.67?

ANS:

A:

B:

C: 519.32 kilowatt-hours

PTS: 3 DIF: medium

3. The number , in hundreds, of soup cans you can sell in a week depends on the price , in dollars, for each can of soup. The relationship is

A: How many cans of soup can you sell in a week if the price is $1.49 per can?

B: Solve for in the equation above to obtain a formula expressing as a function of

C: What is the price per can if you are selling 23.83 hundred cans per week?

ANS:

A: 23.83 hundred cans of soup.

B:

C: $1.49 per can.

PTS: 3 DIF: medium

4. You have 552 pairs of shoes in stock. You sell 35 pairs of shoes each week.

A: Find a formula that gives the number of pairs of shoes in stock after weeks of sales.

B: You need to reorder when stock supplies are reduced to 167 pairs of shoes. After how many weeks will you need to reorder?

ANS:

A:

B: After 11 weeks.

PTS: 2 DIF: medium

5. The chicken sandwiches you are ordering contain a total of 13 grams of trans fat. Fry orders have 10 grams of fat per order.

A: Find a formula that gives the number of grams of fat in your order if you add orders of fries to your sandwich order.

B: How many orders of fries can you make if you wish to limit trans fat in the total order to 43 grams?

ANS: A:

B: 3 orders of fries to accompany your sandwich order.

PTS: 2 DIF: medium

6. The amount , in billions of bushels, that a country is willing to supply depends on the price , in dollars per bushel, of wheat. The relationship is

The demand , in billions of bushels, is the quantity of wheat that customers will buy at a price of dollars per bushel. The relationship is

A: Solve the latter equation for D and so express the demand as a function of the price P

B: The equilibrium price is the price at which supply and demand are the same. Find the equilibrium price.

ANS:

A:

B: 3.9 dollars per bushel

PTS: 2 DIF: medium

7. You are considering jobs that offer similar pay scales.

Pay scale 1: You get a base yearly salary of $10463 plus a commission of 9 percent of yearly sales in dollars.

Pay scale 2: You get a base yearly salary of $13368 plus a commission of 7 percent of yearly sales in dollars.

A: Find a formula for your yearly pay , in dollars, as a function of yearly sales , in dollars, under pay scale 1.

B: Find a formula for your yearly pay , in dollars, as a function of yearly sales , in dollars, under pay scale 2.

C: What amount of yearly sales will result in the same pay under both scales?

ANS:

A:

B:

C: Annual sales of 145,250 dollars will result in the same pay.

PTS: 3 DIF: medium

8. The relationship between Fahrenheit temperature and Celsius temperature is given by

The relationship between the Kelvin temperature scale and the Celsius scale is

A: Find a formula that expresses as a function of

B: Find a formula that expresses as a function of

C: Find a formula that expresses as a function of .

ANS:

A:

B:

C:

PTS: 3 DIF: medium

9. The running speed , in centimeters per second, for certain ants depends on the temperature in degrees Celsius. The relationship is

A: Find a formula that expresses the temperature as a function of the running speed.

B: What is the temperature when the running speed is 6.27 centimeters per second?

ANS:

A:

B: 44.85 degrees Celsius.

PTS: 2 DIF: medium

10. For a certain species of cricket, the temperature , in degrees Fahrenheit, can be estimated from the number of cricket chirps per minute. The relationship is .

A: Find a formula that expresses the number of chirps per minute as a function of the temperature.

B: How many chirps per minute can be expected if the temperature is 56 degrees Fahrenheit?

ANS:

A:

B: 68 chirps per minute

PTS: 2 DIF: medium

11. Between the ages of 7 and 11 years, the weight , in pounds, of a certain girl is given by the formula , where is her age in years.

During the same age range the relationship between her age t and her height , in inches, is given by

A: Find a formula that expresses the girl’s height as a function of her weight. Write your answer in the form , and round the coefficient a of w to two decimal places.

B: Use your answer to part A to determine her height when her weight is 7 pounds.

C: Use your answer to part A to find a formula that expresses the girl’s weight as a function of her height.

ANS:

A:

B: 41.54 inches tall

C:

PTS: 3 DIF: hard

12. The weight of an adult blue whale is related to its length by the formula , where is the length, in feet, and is the weight, in tons.

A: Find a formula that expresses as a function of .

B: How long would the whale need to be to weigh 101 tons?

C: A blue whale is spotted, and its length is estimated to be 84 feet. Use this length to find the whale’s weight.

ANS:

A:

B: 82.67 feet

C: 105.68 tons

PTS: 3 DIF: medium

13. The weight of a baby blue whale over the first 7 months of its life can be calculated using , where is the weight, in tons, and is the age, in months.

A: Find a formula that expresses as a function of .

B: The length , in feet, is given by

Use your answer from part A to find a formula that gives the length as a function of the weight.

C: How long is a baby blue whale if its weight is 19.2 tons?

ANS:

A:

B:

C: 43.58 feet

PTS: 3 DIF: medium

14. An automobile is worth $28857 in 2010. Each year after 2010 the value decreases by $1842.

A: Find a formula that gives the value , in dollars, years after 2010.

B: When will the value of the car be $14121?

ANS:

A:

B: In 2018

PTS: 2 DIF: medium

15. The running speed , in feet per second, of certain animals depends on their length , in inches. The relationship is .

A: Solve this equation for to get an expression that gives the length in terms of the running speed.

B: If an animal 26.32 is inches long, how fast does it run?

C: How long is an animal that runs 26.41 feet per second?

ANS:

A: B: 60.15 feet per second

C: 10.02 inches

PTS: 3 DIF: medium