Precalculus

John W. Coburn

Chapter 8 Systems of Equations and Inequalities - all with Video Answers

Educators

Section 1

Linear Systems in Two Variables with Applications

00:21

Problem 1

Systems that have no solution are called ____ systems.

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00:16

Problem 2

Systems having at least one solution are called ____ systems.

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00:23

Problem 3

If the lines in a system intersect at a single point, the system is said to be ____ and ____.

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00:31

Problem 4

If the lines in a system are coincident, the system is referred to as ____ and ____.

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00:39

The given systems are equivalent. How do we obtain the second system from the first? $\left\{\begin{aligned} \frac{2}{3} x+\frac{1}{2} y &=\frac{5}{3} \\ 0.2 x+0.4 y &=1 \end{aligned}\left\{\begin{array}{l}4 x+3 y=10 \\ 2 x+4 y=10\end{array}\right.\right.$

Mrinal Rana

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01:24

Problem 6

For $\left\{\begin{array}{l}2 x+5 y=8 \\ 3 x+4 y=5\end{array}\right.$ be more efficient, substitution or elimination? Discuss/Explain why.

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00:24

Problem 7

Show the lines in each system would intersect in a single point by writing the equations in slope-intercept form.
$$\left\{\begin{array}{l}7 x-4 y=24 \\4 x+3 y=15\end{array}\right.$$

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00:25

Problem 8

Show the lines in each system would intersect in a single point by writing the equations in slope-intercept form.
$$\left\{\begin{array}{l}0.3 x-0.4 y=2 \\0.5 x+0.2 y=-4\end{array}\right.$$

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00:28

Problem 9

An ordered pair is a solution to an equation if it makes the equation true. Given the graph shown here, determine which equation(s) have the indicated point as a solution. If the point satisfies more than one equation, write the system for which it is a solution.
CAN'T COPY THE GRAPH
$$A$$

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00:21

Problem 10

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00:24

Problem 11

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00:46

Problem 12

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00:35

Problem 13

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00:18

Problem 14

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00:41

Problem 15

Substitute the $x$ - and $y$ -values indicated by the ordered pair to determine if it solves the system.
$$\left\{\begin{array}{c}3 x+y=11 \\-5 x+y=-13\end{array}\right.$$
$$(3,2)$$

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00:58

Problem 16

Substitute the $x$ - and $y$ -values indicated by the ordered pair to determine if it solves the system.
$$\left\{\begin{array}{l}3 x+7 y=-4 \\7 x+8 y=-21\end{array}\right.$$
$$(-6,2)$$

Mrinal Rana

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01:22

Problem 17

Substitute the $x$ - and $y$ -values indicated by the ordered pair to determine if it solves the system.
$$\left\{\begin{aligned}8 x-24 y &=-17 \\12 x+30 y &=2 ;\end{aligned}\left(-\frac{7}{8}, \frac{5}{12}\right)\right.$$

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00:43

Problem 18

Substitute the $x$ - and $y$ -values indicated by the ordered pair to determine if it solves the system.
$$\left\{\begin{array}{l}4 x+15 y=7 \\8 x+21 y=11\end{array},\left(\frac{1}{2}, \frac{1}{3}\right)\right.$$

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01:19

Problem 19

Solve each system by graphing. If the coordinates do not appear to be integers, estimate the solution to the nearest tenth (indicate that your solution is an estimate).
$$\left\{\begin{array}{r}3 x+2 y=12 \\x-y=9\end{array}\right.$$

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00:54

Problem 20

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01:15

Problem 21

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00:54

Problem 22

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01:00

Problem 23

Solve each system using substitution. Write solutions as an ordered pair.
$$\left\{\begin{array}{l}x=5 y-9 \\x-2 y=-6\end{array}\right.$$

Mrinal Rana

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01:06

Problem 24

Solve each system using substitution. Write solutions as an ordered pair.
$$\left\{\begin{array}{l}4 x-5 y=7 \\2 x-5=y\end{array}\right.$$

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01:28

Problem 25

Solve each system using substitution. Write solutions as an ordered pair.
$$\left\{\begin{array}{l}y=\frac{2}{3} x-7 \\3 x-2 y=19\end{array}\right.$$

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01:15

Problem 26

Solve each system using substitution. Write solutions as an ordered pair.
$$\left\{\begin{array}{l}2 x-y=6 \\y=\frac{3}{4} x-1\end{array}\right.$$

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01:31

Problem 27

Identify the equation and variable that makes the substitution method easiest to use. Then solve the system.
$$\left\{\begin{array}{l}3 x-4 y=24 \\5 x+y=17\end{array}\right.$$

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01:14

Problem 28

Identify the equation and variable that makes the substitution method easiest to use. Then solve the system.
$$\left\{\begin{array}{r}3 x+2 y=19 \\x-4 y=-3\end{array}\right.$$

Mrinal Rana

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01:41

Problem 29

Identify the equation and variable that makes the substitution method easiest to use. Then solve the system.
$$\left\{\begin{aligned}0.7 x+2 y &=5 \\x-1.4 y &=11.4\end{aligned}\right.$$

Mrinal Rana

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01:45

Problem 30

Identify the equation and variable that makes the substitution method easiest to use. Then solve the system.
$$\left\{\begin{array}{l}0.8 x+y=7.4 \\0.6 x+1.5 y=9.3\end{array}\right.$$

Mrinal Rana

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01:16

Problem 31

Identify the equation and variable that makes the substitution method easiest to use. Then solve the system.
$$\left\{\begin{array}{c}5 x-6 y=2 \\x+2 y=6\end{array}\right.$$

Mrinal Rana

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01:37

Problem 32

Identify the equation and variable that makes the substitution method easiest to use. Then solve the system.
$$\left\{\begin{array}{l}2 x+5 y=5 \\8 x-y=6\end{array}\right.$$

Mrinal Rana

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00:52

Problem 33

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}2 x-4 y=10 \\3 x+4 y=5\end{array}\right.$$

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00:36

Problem 34

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{r}-x+5 y=8 \\x+2 y=6\end{array}\right.$$

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00:49

Problem 35

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}4 x-3 y=1 \\3 y=-5 x-19\end{array}\right.$$

Mrinal Rana

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00:44

Problem 36

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}5 y-3 x=-5 \\3 x+2 y=19\end{array}\right.$$

Mrinal Rana

Numerade Educator

01:16

Problem 37

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}2 x=-3 y+17 \\4 x-5 y=12\end{array}\right.$$

Mrinal Rana

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01:11

Problem 38

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}2 y=5 x+2 \\-4 x=17-6 y\end{array}\right.$$

Mrinal Rana

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01:32

Problem 39

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}0.5 x+0.4 y=0.2 \\0.3 y=1.3+0.2 x\end{array}\right.$$

Mrinal Rana

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01:51

Problem 40

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}0.2 x+0.3 y=0.8 \\0.3 x+0.4 y=1.3\end{array}\right.$$

Mrinal Rana

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01:49

Problem 41

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{aligned}0.32 m-0.12 n &=-1.44 \\-0.24 m+0.08 n &=1.04\end{aligned}\right.$$

Mrinal Rana

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01:34

Problem 42

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{aligned}0.06 g-0.35 h &=-0.67 \\-0.12 g+0.25 h &=0.44\end{aligned}\right.$$

Mrinal Rana

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01:00

Problem 43

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{aligned}-\frac{1}{6} u+\frac{1}{4} v &=4 \\\frac{1}{2} u-\frac{2}{3} v &=-11\end{aligned}\right.$$

Mrinal Rana

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01:13

Problem 44

Solve using elimination. In some cases, the system must first be written in standard form.
$$\left\{\begin{array}{l}\frac{3}{4} x+\frac{1}{3} y=-2 \\\frac{3}{2} x+\frac{1}{3} y=3\end{array}\right.$$

Mrinal Rana

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02:03

Problem 45

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{aligned}4 x+\frac{3}{4} y &=14 \\-9 x+\frac{5}{8} y &=-13\end{aligned}\right.$$

Mrinal Rana

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01:14

Problem 46

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}\frac{2}{3} x+y=2 \\2 y=\frac{5}{6} x-9\end{array}\right.$$

Mrinal Rana

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00:44

Problem 47

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}0.2 y=0.3 x+4 \\0.6 x-0.4 y=-1\end{array}\right.$$

Mrinal Rana

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00:36

Problem 48

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}1.2 x+0.4 y=5 \\0.5 y=-1.5 x+2\end{array}\right.$$

Mrinal Rana

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00:40

Problem 49

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}6 x-22=-y \\3 x+\frac{1}{2} y=11\end{array}\right.$$

Mrinal Rana

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00:48

Problem 50

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{aligned}15-5 y &=-9 x \\-3 x+\frac{5}{3} y &=5\end{aligned}\right.$$

Mrinal Rana

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01:38

Problem 51

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}-10 x+35 y=-5 \\y=0.25 x\end{array}\right.$$

Mrinal Rana

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01:18

Problem 52

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}2 x+3 y=4 \\x=-2.5 y\end{array}\right.$$

Mrinal Rana

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01:22

Problem 53

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}7 a+b=-25 \\2 a-5 b=14\end{array}\right.$$

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00:37

Problem 54

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{aligned}-2 m+3 n &=-1 \\5 m-6 n &=4\end{aligned}\right.$$

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00:56

Problem 55

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}4 a=2-3 b \\6 b+2 a=7\end{array}\right.$$

Mrinal Rana

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01:15

Problem 56

Solve using any method and identify the system as consistent, inconsistent, or dependent.
$$\left\{\begin{array}{l}3 p-2 q=4 \\9 p+4 q=-3\end{array}\right.$$

Mrinal Rana

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00:56

Problem 57

The substitution method can be used for like variables or for like expressions. Solve the following systems, using the expression common to both equations (do not solve for $x$ or $y$ alone).
$$\left\{\begin{array}{r}2 x+4 y=6 \\x+12=4 y\end{array}\right.$$

Mrinal Rana

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01:06

Problem 58

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00:56

Problem 59

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00:47

Problem 60

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00:48

Problem 61

$$\text { Uniform motion with current: }\left\{\begin{array}{c}(R+C) T_{1}=D_{1} \\(R-C) T_{2}=D_{2}\end{array}\right.$$
The formula shown can be used to solve uniform motion problems involving a current, where $D$ represents distance traveled, $R$ is the rate of the object with no current, $C$ is the speed of the current, and $T$ is the time. Chan-Li rows 9 mi up river (against the current) in 3 hr. It only took him 1 hr to row 5 mi downstream (with the current). How fast was the current? How fast can he row in still water?

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Problem 62

Fahrenheit and Celsius temperatures:
$\left\{\begin{array}{ll}y=3 x+32 & \text { 'F } \\ y=\frac{5}{9}(x-32) & \text { "C }\end{array}\right.$
Many people are familiar with temperature measurement in degrees Celsius and degrees Fahrenheit, but few realize that the equations are linear and there is one temperature at which the two scales agree. Solve the system using the method of your choice and find this temperature.

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02:46

Problem 63

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
At a recent production of $A$ Comedy of Errors, the Community Theater brought in a total of 30,495 dollars in revenue. If adult tickets were 9 dollars and children's tickets were 6.50 dollars, how many tickets of each type were sold if 3800 tickets in all were sold?

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Problem 64

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
A dietician needs to mix 10 gal of milk that is $2 \frac{10}{2} \%$ milk fat for the day's rounds. He has some milk that is $4 \%$ milk fat and some that is $1 \frac{1}{2} \%$ milk fat. How much of each should be used?

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02:16

Problem 65

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
Cherokee just filled both of the family vehicles at a service station. The total cost for 20 gal of regular unleaded and 17 gal of premium unleaded was 144.89 dollars .The premium gas was 0.10 dollar more per gallon than the regular gas. Find the price per gallon for each type of gasoline.

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Problem 66

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
As a cleaning agent, a solution that is $24 \%$ vinegar is often used. How much pure $(100 \%)$ vinegar and $5 \%$ vinegar must be mixed to obtain 50 oz of a $24 \%$ solution?

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02:27

Problem 67

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
A wealthy alumnus donated 10,000 dollars to his alma mater. The college used the funds to make a loan to a science major at $7 \%$ interest and a loan to a nursing student at $6 \%$ interest. That year the college earned 635 dollars in interest. How much was loaned to each student?

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02:24

Problem 68

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
A total of 12,000 dollars is invested in two municipal bonds, one paying $10.5 \%$ and the other $12 \%$ simple interest. Last year the annual interest earned on the two investments was 1335 dollars. How much was invested at each rate?

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01:03

Problem 69

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
Bryan has been doing odd jobs around the house, trying to earn enough money to buy a new Dirt-Surfer@. He saves all quarters and dimes in his piggy bank, while he places all nickels and pennies in a drawer to spend. So far, he has
225 coins in the piggy bank, worth a total of 45.00 dollars . How many of the coins are quarters? How many are dimes?

Mrinal Rana

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01:36

Problem 70

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
In $1990,$ Molly attended a coin auction and purchased some rare "Flowing Hair" fifty-cent pieces, and a number of very rare twocent pieces from the Civil War Era. If she bought 47 coins with a face value of 10.06 dollars, how many of each denomination did she buy?

Mrinal Rana

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01:18

Problem 71

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
Dave and his sons run a lawn service, which includes mowing, edging, trimming, and aerating a lawn. His fixed cost includes insurance, his salary, and monthly payments on equipment, and amounts to 4000 dollars/mo. The variable costs include gas, oil, hourly wages for his employees, and miscellaneous expenses, which run about 75 dollars per lawn. The average charge for full service lawn care is 115 dollars per visit. Do a breakeven analysis to (a) determine how many lawns Dave must service each month to break even and (b) the revenue required to break even.

Mrinal Rana

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01:40

Problem 72

Solve each application by modeling the situation with a linear system. Be sure to clearly indicate what each variable represents.
Due to high market demand, a manufacturer decides to introduce a new line of mini-microwave ovens for personal and office use. By using existing factory space and retraining some employees, fixed costs are estimated at 8400 dollars/mo. The components to assemble and test each microwave are expected to run 45 dollars per unit. If market research shows consumers are willing to pay at least 69 dollars for this product, find (a) how many units must be made and sold each month to break even and (b) the revenue required to break even.

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05:28

Problem 73

One area where the law of supply and demand is clearly at work is farm commodities. Both growers and consumers watch this relationship closely, and use data collected by government agencies to track the relationship and make adjustments, as when a farmer decides to convert a large portion of her farmland from corn to soybeans to improve profits. Suppose that for $x$ billion bushels of soybeans, supply is modeled by $y=1.5 x+3,$ where $y$ is the current market price (in dollars per bushel). The related demand equation might be $y=-2.20 x+12 .$ (a) How many billion bushels will be supplied at a market price of 5.40 dollars? What will the demand be at this price? Is supply less than demand? (b) How many billion bushels will be supplied at a market price of 7.05 dollars? What will the demand be at this price? Is demand less than supply? (c) To the nearest cent, at what price does the market reach equilibrium? How many bushels are being supplied/demanded?

Mrinal Rana

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03:24

Problem 74

Market research has indicated that by $2010,$ sales of MP3 portables will mushroom into a 70 billion dollar market. With a market this large, competition is often fierce- -with suppliers fighting to earn and hold market shares. For $x$ million MP3 players sold, supply is modeled by $y=10.5 x+25,$ where $y$ is the current market price (in dollars). The related demand equation might be $y=-5.20 x+140 .$ (a) How many million MP3 players will be supplied at a market price of 88 dollars ? What will the demand be at this price? Is supply less than demand? (b) How many million MP3 players will be supplied at a market price of 114 dollars ? What will the demand be at this price? Is demand less than supply? (c) To the nearest cent, at what price does the market reach equilibrium? How many units are being supplied/demanded?

Mrinal Rana

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01:44

Problem 75

On a recent camping trip, it took Molly and Sharon 2 hr to row 4 mi upstream from the drop in point to the campsite. After a leisurely weekend of camping, fishing, and relaxation, they rowed back downstream to the drop in point in just 30 min. Use this information to find (a) the speed of the current and (b) the speed Sharon and Molly would be rowing in still water.

Mrinal Rana

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01:55

Problem 76

A luxury ship is taking a Caribbean cruise from Caracas, Venezuela, to just off the coast of Belize City on the Yucatan Peninsula, a distance of 1435 mi. En route they encounter the Caribbean Current, which flows to the northwest, parallel to the coastline. From Caracas to the Belize coast, the trip took 70 hr. After a few days of fun in the sun, the ship leaves for Caracas, with the return trip taking 82 hr. Use this information to find (a) the speed of the Caribbean Current and (b) the cruising speed of the ship.

Mrinal Rana

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02:23

Problem 77

As part of an algebra field trip, Jason takes his class to the airport to use their moving walkways for a demonstration. The class measures the longest walkway, which turns out to be $256 \mathrm{ft}$ long. Using a stop watch, Jason shows it takes him just 32 sec to complete the walk going in the same direction as the walkway. Walking in a direction opposite the walkway, it takes him $320 \mathrm{sec}-10$ times as long! The next day in class, Jason hands out a two-question quiz: (1) What was the speed of the walkway in feet per second? (2) What is my (Jason's) normal walking speed? Create the answer key for this quiz.

Mrinal Rana

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02:17

Problem 78

The American Racing Pigeon Union often sponsors opportunities for owners to fly their birds in friendly competitions. During a recent competition, Steve's birds were liberated in Topeka, Kansas, and headed almost due north to their loft in Sioux Falls, South Dakota, a distance of 308 mi. During the flight, they encountered a steady wind from the north and the trip took 4.4 hr. The next month, Steve took his birds to a competition in Grand Forks, North Dakota, with the birds heading almost due south to home, also a distance of 308 mi. This time the birds were aided by the same wind from the north, and the trip took only 3.5 hr. Use this information to (a) find the racing speed of Steve's birds and (b) find the speed of the wind.

Mrinal Rana

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01:50

Problem 79

If you sum the year that the Declaration of Independence was signed and the year that the Civil War ended, you get $3641 .$ There are 89 yr that separate the two events. What year was the Declaration signed? What year did the Civil War end?

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01:58

Problem 80

When it was first constructed in $1889,$ the Eiffel Tower in Paris, France, was the tallest structure in the world. In $1975,$ the $\mathrm{CN}$ Tower in Toronto, Canada, became the world's tallest structure. The CN Tower is 153 ft less than twice the height of the Eiffel Tower, and the sum of their heights is 2799 ft. How tall is each tower?

Mrinal Rana

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01:45

Problem 81

In the South Pacific, the island nations of Tahiti and Tonga have a combined land area of $692 \mathrm{mi}^{2}$. Tahiti's land area is $112 \mathrm{mi}^{2}$ more than Tonga's. What is the land area of each island group?

Mrinal Rana

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01:16

Problem 82

On a cold winter night, in the lobby of a beautiful hotel in Sante Fe, New Mexico, Marc and Klay just barely beat John and Steve in a close game of Trumps. If the sum of the team scores was 990 points, and there was a 12 -point margin of victory, what was the final score?

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Numerade Educator

Problem 83

Answer using observations only-no calculations. Is the given system consistent/independent, consistent/dependent, or inconsistent? Explain/Discuss your answer. $\left\{\begin{array}{l}y=5 x+2 \\ y=5.01 x+1.9\end{array}\right.$

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01:05

Problem 84

Federal income tax reform has been a hot political topic for many years. Suppose tax plan A calls for a flat tax of $20 \%$ tax on all income (no deductions or loopholes). Tax plan B requires taxpayers to pay 5000 dollars plus $10 \%$ of all income. For what income level do both plans require the same tax?

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04:05

Problem 85

Suppose a certain amount of money was invested at $6 \%$ per year, and another amount at $8.5 \%$ per year, with a total return of 1250 dollars. If the amounts invested at each rate were switched, the yearly income would have been 1375 dollars. To the nearest whole dollar, how much was invested at each rate?

Mrinal Rana

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