Precalculus with Limits : A Graphing Approach

Ron Larson, Robert Hostetler, Bruce H. Edwards

Chapter 7

Linear Systems and Matrices - all with Video Answers

Educators

Section 1

Solving Systems of Equations

Problem 1

In Exercises $1-4,$ determine whether each ordered pair is a solution of the system of equations.
$$
\left\{\begin{array}{lll}{4 x-y=1} & {\text { (a) }(0,-3)} & {\text { (b) }(-1,-5)} \\ {6 x+y=-6} & {\text { (c) }\left(-\frac{3}{2}, 3\right)} & {\text { (d) }\left(-\frac{1}{2},-3\right)}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 2

In Exercises $1-4,$ determine whether each ordered pair is a solution of the system of equations.
$$
\left\{\begin{array}{lll}{4 x^{2}+y=3} & {\text { (a) }(2,-13)} & {\text { (b) }(-2,-9)} \\ {-x-y=11} & {\text { (c) }\left(-\frac{3}{2}, 6\right)} & {\text { (d) }\left(-\frac{7}{4},-\frac{37}{4}\right)}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 3

In Exercises $1-4,$ determine whether each ordered pair is a solution of the system of equations.
$$
\left\{\begin{array}{ccc}{y=-2 e^{x}} & {(\text { a) }(-2,0)} & {\text { (b) }(0,-2)} \\ {3 x-y=2} & {\text { (c) }(0,-3)} & {\text { (d) }(-1,-5)}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 4

In Exercises $1-4,$ determine whether each ordered pair is a solution of the system of equations.
$$
\left\{\begin{array}{rlrl}{-\log _{10} x+3} & {=y} & {} & {\text { (a) }(100,1)} & {\text { (b) }(10,2)} \\ {\frac{1}{9} x+y} & {=\frac{28}{9}} && {\text { (c) }(1,3)} & {\text { (d) }(1,1)}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 5

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{array}{c}{2 x+y=6} \\ {-x+y=0}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 6

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{array}{l}{x-y=-4} \\ {x+2 y=5}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 7

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{array}{c}{x-y=-4} \\ {x^{2}-y=-2}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 8

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{aligned}-2 x+y &=-5 \\ x^{2}+y^{2} &=25 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 9

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{aligned} 3 x+y &=2 \\ x^{3}-2+y &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 10

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{aligned} x+y &=0 \\ x^{3}-5 x-y &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 11

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{aligned}-\frac{7}{2} x-y &=-18 \\ 8 x^{2}-2 y^{3} &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 12

In Exercises $5-12,$ solve the system by the method of substitution. Check your solution graphically.
$$
\left\{\begin{array}{l}{y=x^{3}-3 x^{2}+4} \\ {y=-2 x+4}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 13

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{c}{x-y=0} \\ {5 x-3 y=10}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 14

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{c}{x+2 y=1} \\ {5 x-4 y=-23}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 15

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{l}{2 x-y+2=0} \\ {4 x+y-5=0}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 16

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{r}{6 x-3 y-4=0} \\ {x+2 y-4=0}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 17

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{l}{1.5 x+0.8 y=2.3} \\ {0.3 x-0.2 y=0.1}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 18

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{l}{0.5 x+3.2 y=9.0} \\ {0.2 x-1.6 y=-3.6}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 19

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{aligned} \frac{1}{5} x+\frac{1}{2} y &=8 \\ x+y &=20 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 20

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{l}{\frac{1}{2} x+\frac{3}{4} y=10} \\ {\frac{3}{4} x-y=4}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 21

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{l}{-\frac{5}{3} x+y=5} \\ {-5 x+3 y=6}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 22

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{aligned}-\frac{2}{3} x+y &=2 \\ 2 x-3 y &=6 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 23

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{aligned} x^{2}-2 x+y &=8 \\ x-y &=-2 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 24

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{aligned} 2 x^{2}-2 x-y &=14 \\ 2 x-y &=-2 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 25

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{c}{2 x^{2}-y=1} \\ {x-y=2}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 26

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{c}{2 x^{2}+y=3} \\ {x+y=4}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 27

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{aligned} x^{3}-y &=0 \\ x-y &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 28

In Exercises $13-28,$ solve the system by the method of substitution. Use a graphing utility to verify your results.
$$
\left\{\begin{array}{l}{y=-x} \\ {y=x^{3}+3 x^{2}+2 x}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 29

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{array}{c}{-x+2 y=2} \\ {3 x+y=15}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 30

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{array}{c}{x+y=0} \\ {3 x-2 y=10}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 31

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{aligned} x-3 y &=-2 \\ 5 x+3 y &=17 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 32

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{aligned}-x+2 y &=1 \\ x-y &=2 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 33

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{array}{l}{x^{2}+y=1} \\ {x+y=2}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 34

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{array}{c}{x^{2}-y=4} \\ {x-y=2}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 35

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{aligned}-x+y &=3 \\ x^{2}+y^{2}-6 x-27 &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 36

In Exercises $29-36,$ solve the system graphically. Verify your solutions algebraically.
$$
\left\{\begin{aligned} y^{2}-4 x+11 &=0 \\-\frac{1}{2} x+y &=-\frac{1}{2} \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 37

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{c}{7 x+8 y=24} \\ {x-8 y=8}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 38

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{c}{x-y=0} \\ {5 x-2 y=6}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 39

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{l}{x-y^{2}=-1} \\ {x-y=5}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 40

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{l}{x-y^{2}=-2} \\ {x-2 y=6}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 41

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{aligned} x^{2}+y^{2} &=8 \\ y &=x^{2} \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 42

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{aligned} x^{2}+y^{2} &=25 \\(x-8)^{2}+y^{2} &=41 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 43

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{aligned} y &=e^{x} \\ x-y+1 &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 44

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{aligned} y &=-4 e^{-x} \\ y+3 x+8 &=0 \end{aligned}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 45

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{aligned} x+2 y &=8 \\ y &=2+\ln x \end{aligned}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 46

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{aligned} y &=-2+\ln (x-1) \\ 3 y+2 x &=9 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 47

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{l}{y=\sqrt{x}+4} \\ {y=2 x+1}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 48

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{c}{x-y=3} \\ {\sqrt{x}-y=1}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 49

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x^{2}-8 y=104}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 50

In Exercises 37–50, use a graphing utility to approximate all points of intersection of the graph of the system of equations. Round your results to three decimal places. Verify your solutions by checking them in the original system.
$$
\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {2 x^{2}-y=2}\end{array}\right.
$$

Wali Jan

Numerade Educator

Problem 51

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{2 x-y=0} \\ {x^{2}-y=-1}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 52

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{c}{x+y=4} \\ {x^{2}+y=2}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 53

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{c}{3 x-7 y=-6} \\ {x^{2}-y^{2}=4}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 54

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {2 x+y=10}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 55

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{x^{2}+y^{2}=1} \\ {x+y=4}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 56

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x-y=5}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 57

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{y=2 x+1} \\ {y=\sqrt{x+2}}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 58

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{y=2 x-1} \\ {y=\sqrt{x+1}}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 59

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{y-e^{-x}=1} \\ {y-\ln x=3}\end{array}\right.
$$

Charles Carter

Numerade Educator

Problem 60

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{aligned} 2 \ln x+y &=4 \\ e^{x}-y &=0 \end{aligned}\right.
$$

Wali Jan

Numerade Educator

Problem 61

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{y=x^{3}-2 x^{2}+1} \\ {y=1-x^{2}}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 62

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{array}{l}{y=x^{3}-2 x^{2}+x-1} \\ {y=-x^{2}+3 x-1}\end{array}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 63

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{aligned} x y-1 &=0 \\ 2 x-4 y+7 &=0 \end{aligned}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 64

In Exercises $51-64$ , solve the system graphically or algebraically. Explain your choice of method.
$$
\left\{\begin{aligned} x y-2 &=0 \\ 3 x-2 y+4 &=0 \end{aligned}\right.
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 65

Break-Even Analysis In Exercises $65-68,$ use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales $x$ necessary to break even $(R=C)$ and the corresponding revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
$$
C=8650 x+250,000 \quad R=9950 x
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 66

Break-Even Analysis In Exercises $65-68,$ use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales $x$ necessary to break even $(R=C)$ and the corresponding revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
$$
C=2.65 x+350,000 \quad R=4.15 x
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 67

Break-Even Analysis In Exercises $65-68,$ use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales $x$ necessary to break even $(R=C)$ and the corresponding revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
$$
C=5.5 \sqrt{x}+10,000 \quad R=3.29 x
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 68

Break-Even Analysis In Exercises $65-68,$ use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales $x$ necessary to break even $(R=C)$ and the corresponding revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
$$
C=7.8 \sqrt{x}+18,500 \quad R=12.84 x
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 69

DVD Rentals The daily DVD rentals of a newly released animated film and a newly released horror film from a movie rental store can be modeled by the equations
$$ \left\{\begin{array}{ll}{N=360-24 x} & {\text { Animated film }} \\ {N=24+18 x} & {\text { Horror film }}\end{array}\right.
$$
where $N$ is the number of $\mathrm{DVDs}$ rented and $x$ represents the week, with $x=1$ corresponding to the first week of release.
(a) Use the table feature of a graphing utility to find the numbers of rentals of each movie for each of the first 12 weeks of release.
(b) Use the results of part (a) to determine the solution to the system of equations.
(c) Solve the system of equations algebraically.
(d) Compare your results from parts (b) and (c).
(e) Interpret the results in the context of the situation.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 70

Sports The points scored during each of the first 12 games by two players on a girl's high school basketball team can be modeled by the equations
$$
\left\{\begin{array}{l}{P_{S}=24-2 x} \\ {P_{P}=12+2 x}\end{array}\right.
$$
where $P$ represents the points scored by each player and $x$ represents the number of games played, with $x=1$ corresponding to the first game.
(a) Use the table feature of a graphing utility to find the numbers of points scored by each player for each of the first 12 games.
(b) Use the results of part (a) to determine the solution to the system of equations.
(c) Solve the system of equations algebraically.
(d) Compare your results from parts (b) and (c).
(e) Interpret the results in the context of the situation.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 71

Break-Even Analysis A small software company invests $\$ 16,000$ to produce a software package that will sell for $\$ 55.95$ . Each unit can be produced for $\$ 35.45 .$
(a) Write the cost and revenue functions for $x$ units produced and sold.
(b) Use a graphing utility to graph the cost and revenue functions in the same viewing window. Use the graph to approximate the number of units that must be sold to break even, and verify the result algebraically.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 72

Break-Even Analysis A small fast food restaurant invests $\$ 5000$ to produce a new food item that will sell for $\$ 3.49$ . Each item can be produced for $\$ 2.16$ .
(a) Write the cost and revenue functions for $x$ items produced and sold.
(b) Use a graphing utility to graph the cost and revenue functions in the same viewing window. Use the graph to approximate the number of items that must be sold to break even, and verify the result algebraically.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 73

Choice of Two Jobs You are offered two different jobs selling dental supplies. One company offers a straight commission of 6$\%$ of sales. The other company offers a salary of $\$ 350$ per week plus 3$\%$ of sales. How much would you have to sell in a week in order to make the straight commission offer the better offer?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 74

Choice of Two Jobs You are offered two jobs selling college textbooks. One company offers an annual salary of $\$ 25,000$ plus a year-end bonus of 1$\%$ of your total sales. The other company offers an annual salary of $\$ 20,000$ plus a year-end bonus of 2$\%$ of your total sales. How much would you have to sell in a year to make the second offer the better offer?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 75

Investment A total of $\$ 20,000$ is invested in two funds paying 6.5$\%$ and 8.5$\%$ simple interest. The 6.5$\%$ investment has a lower risk. The investor wants a yearly interest check of $\$ 1600$ from the investments.
(a) Write a system of equations in which one equation represents the total amount invested and the other equation represents the $\$ 1600$ required in interest. Let $x$ and $y$ represent the amounts invested at 6.5$\%$ and 8.5$\%$ respectively.
(b) Use a graphing utility to graph the two equations in the same viewing window. As the amount invested at 6.5$\%$ increases, how does the amount invested at 8.5$\%$ change? How does the amount of interest change? Explain.
(c) What amount should be invested at 6.5$\%$ to meet the requirement of $\$ 1600$ per year in interest?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 76

Log Volume You are offered two different rules for estimating the number of board feet in a 16 -foot log. (A board foot is a unit of measure for lumber equal to a board 1 foot square and 1 inch thick.) One rule is the Doyle Log Rule and is modeled by
$$V=(D-4)^{2}, \quad 5 \leq D \leq 40$$
and the other rule is the Scribner Log Rule and is modeled by
$$V=0.79 D^{2}-2 D-4, \quad 5 \leq D \leq 40$$
where $D$ is the diameter (in inches) of the log and $V$ is its volume in (board feet).
(a) Use a graphing utility to graph the two log rules in the same viewing window.
(b) For what diameter do the two rules agree?
(c) You are selling large logs by the board foot. Which rule would you use? Explain your reasoning.

AG

Ankit Gupta

Numerade Educator

Problem 77

Population The populations (in thousands) of Missouri $M$ and Tennessee $T$ from 1990 to 2004 can by modeled by the system
$$
\left\{\begin{array}{l}{M=47.4 t+5104} \\ {T=76.5 t+4875}\end{array}\right.
$$
where $t$ is the year, with $t=0$ corresponding to $1990 .$ (Source: U.S. Census Bureau)
(a) Record in a table the populations of the two states for the years $1990,1994,1998,2002,2006,$ and 2010 .
(b) According to the table, over what period of time does the population of Tennessee exceed that of Missouri?
(c) Use a graphing utility to graph the models in the same viewing window. Estimate the point of intersection of the models.
(d) Find the point of intersection algebraically.
(e) Summarize your findings of parts (b) through (d).

Mrinal Rana

Numerade Educator

Problem 78

Tuition The table shows the average costs (in dollars) of one year's tuition for public and private universities in the United States from 2000 to 2004 . (Source: U.S, National Center for Education Statistics)
(a) Use the regression feature of a graphing utility to find a quadratic model $T_{\text {public}}$ for tuition at public universities and a linear model $T_{\text {private}}$ for tuition at private universities. Let $x$ represent the year, with $x=0$ corresponding to 2000 .
(b) Use a graphing utility to graph the models with the original data in the same viewing window.
(c) Use the graph in part (b) to determine the year after 2004 in which tuition at public universities will exceed tuition at private universities.
(d) Algebraically determine the year in which tuition at public universities will exceed tuition at private
universities.
(e) Compare your results from parts (c) and (d).

Noah Musser

Numerade Educator

Problem 79

Geometry In Exercises 79 and 80, find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 30 meters and the length is 3 meters greater than the width.

Wali Jan

Numerade Educator

Problem 80

Geometry In Exercises 79 and 80, find the dimensions of the rectangle meeting the specified conditions.
The perimeter is 280 centimeters and the width is 20 centimeters less than the length.

Wali Jan

Numerade Educator

Problem 81

Geometry What are the dimensions of a rectangular tract of land if its perimeter is 40 miles and its area is 96 square miles?

Wali Jan

Numerade Educator

Problem 82

Geometry What are the dimensions of an isosceles right triangle with a two-inch hypotenuse and an area of 1 square inch?

Ivan Kochetkov

Numerade Educator

Problem 83

True or False? In Exercises 83 and $84,$ determine whether the statement is true or false. Justify your answer.
In order to solve a system of equations by substitution, you must always solve for $y$ in one of the two equations and then back-substitute.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 84

True or False? In Exercises 83 and $84,$ determine whether the statement is true or false. Justify your answer.
If a system consists of a parabola and a circle, then it can have at most two solutions.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 85

Think About It When solving a system of equations by substitution, how do you recognize that the system has no solution?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 86

Writing Write a brief paragraph describing any advantages of substitution over the graphical method of
solving a system of equations.

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 87

Exploration Find the equations of lines whose graphs intersect the graph of the parabola $y=x^{2}$ at (a) two points, (b) one point, and (c) no points. (There are many correct answers.)

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 88

Exploration Create systems of two linear equations in two variables that have (a) no solution, (b) one distinct solution, and (c) infinitely many solutions. (There are many correct answers.)

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 89

Exploration Create a system of linear equations in two variables that has the solution $(2,-1)$ as its only solution. (There are many correct answers.)

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 90

Conjecture Consider the system of equations.
$$\left\{\begin{array}{l}{y=b^{x}} \\ {y=x^{b}}\end{array}\right.$$
(a) Use a graphing utility to graph the system of equations for $b=2$ and $b=4$
(b) For a fixed value of $b>1$ , make a conjecture about the number of points of intersection of the graphs in part (a).

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 91

In Exercises $91-96$ , find the general form of the equation of the line passing through the two points.
$$
(-2,7),(5,5)
$$

Wali Jan

Numerade Educator

Problem 92

In Exercises $91-96$ , find the general form of the equation of the line passing through the two points.
$$
(3,4),(10,6)
$$

Wali Jan

Numerade Educator

Problem 93

In Exercises $91-96$ , find the general form of the equation of the line passing through the two points.
$$
(6,3),(10,3)
$$

Wali Jan

Numerade Educator

Problem 94

In Exercises $91-96$ , find the general form of the equation of the line passing through the two points.
$$
(4,-2),(4,5)
$$

Wali Jan

Numerade Educator

Problem 95

In Exercises $91-96$ , find the general form of the equation of the line passing through the two points.
$$
\left(\frac{3}{5}, 0\right),(4,6)
$$

Wali Jan

Numerade Educator

Problem 96

In Exercises $91-96$ , find the general form of the equation of the line passing through the two points.
$$
\left(-\frac{7}{3}, 8\right),\left(\frac{5}{2}, \frac{1}{2}\right)
$$

Wali Jan

Numerade Educator

Problem 97

In Exercises $97-102,$ find the domain of the function and identify any horizontal or vertical asymptotes.
$$
f(x)=\frac{5}{x-6}
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 98

In Exercises $97-102,$ find the domain of the function and identify any horizontal or vertical asymptotes.
$$
f(x)=\frac{2 x-7}{3 x+2}
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 99

In Exercises $97-102,$ find the domain of the function and identify any horizontal or vertical asymptotes.
$$
f(x)=\frac{x^{2}+2}{x^{2}-16}
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 100

In Exercises $97-102,$ find the domain of the function and identify any horizontal or vertical asymptotes.
$$
f(x)=3-\frac{2}{x^{2}}
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 101

In Exercises $97-102,$ find the domain of the function and identify any horizontal or vertical asymptotes.
$$
f(x)=\frac{x+1}{x^{2}+1}
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 102

In Exercises $97-102,$ find the domain of the function and identify any horizontal or vertical asymptotes.
$$
f(x)=\frac{x-4}{x^{2}+16}
$$

Dharmendra Jain

Dharmendra Jain

Numerade Educator