Precalculus with Limits (2010)

Educators

Problem 1

A set of two or more equations in two or more variables is called a ________ of ________.

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Problem 2

A ________ of a system of equations is an ordered pair that satisfies each equation in the system.

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Problem 3

Finding the set of all solutions to a system of equations is called ________ the system of equations.

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Problem 4

The first step in solving a system of equations by the method of ________ is to solve one of the equations
for one variable in terms of the other variable.

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Problem 5

Graphically, the solution of a system of two equations is the ________ of ________ of the graphs of the two equations.

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Problem 6

In business applications, the point at which the revenue equals costs is called the ________ point.

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Problem 7

In Exercises 7–10, determine whether each ordered pair is a solution of the system of equations.
$\left\{\begin{array}{l}{2 x-y=4} \\ {8 x+y=-9}\end{array}\right.\begin{array}{ll}{\text { (a) }(0,-4)} & {\text { (b) }(-2,7)} \\ {\text { (c) }\left(\frac{3}{2},-1\right)} & {\text { (d) }\left(-\frac{1}{2},-5\right)}\end{array}$

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Problem 8

In Exercises 7–10, determine whether each ordered pair is a solution of the system of equations.
$\left\{\begin{array}{l}{4 x^{2}+y=3} \\ {-x-y=11}\end{array}\right.\begin{array}{l}{\text { (a) }(2,-13) \quad \text { (b) }(2,-9)} \\ {\text { (c) }\left(-\frac{3}{2},-\frac{31}{3}\right) \text { (d) }\left(-\frac{2}{4},-\frac{37}{4}\right)}\end{array}$

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Problem 9

In Exercises 7–10, determine whether each ordered pair is a solution of the system of equations.
\left\{\begin{aligned} y &=-4 e^{x} \\ 7 x-y &=4 \end{aligned}\right.\begin{array}{ll}{\text { (a) }(-4,0)} & {\text { (b) }(0,-4)} \\ {\text { (c) }(0,-2)} & {\text { (d) }(-1,-3)}\end{array}

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Problem 10

In Exercises 7–10, determine whether each ordered pair is a solution of the system of equations.
\left\{\begin{aligned}-\log x+3 &=y \\ \frac{1}{9} x+y &=\frac{28}{9} \end{aligned}\right.\begin{array}{ll}{\text { (a) }\left(9, \frac{37}{9}\right)} & {\text { (b) }(10,2)} \\ {\text { (c) }(1,3)} & {\text { (d) }(2,4)}\end{array}

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Problem 11

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{2 x+y=6} \\ {-x+y=0}\end{array}\right.$$

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Problem 12

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{x-4 y=-11} \\ {x+3 y=3}\end{array}\right.$$

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Problem 13

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{x-y=-4} \\ {x^{2}-y=-2}\end{array}\right.$$

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Problem 14

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
\left\{\begin{aligned} 3 x+y &=2 \\ x^{3}-2+y &=0 \end{aligned}\right.

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Problem 15

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{-\frac{1}{2} x+y=-\frac{5}{2}} \\ {x^{2}+y^{2}=25}\end{array}\right.$$

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Problem 16

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
\left\{\begin{aligned} x+y &=0 \\ x^{3}-5 x-y &=0 \end{aligned}\right.

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Problem 17

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
\left\{\begin{aligned} x^{2}+y &=0 \\ x^{2}-4 x-y &=0 \end{aligned}\right.

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Problem 18

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{y=-2 x^{2}+2} \\ {y=2\left(x^{4}-2 x^{2}+1\right)}\end{array}\right.$$

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Problem 19

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{y=x^{3}-3 x^{2}+1} \\ {y=x^{2}-3 x+1}\end{array}\right.$$

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Problem 20

In Exercises $11-20,$ solve the system by the method of substitution. Check your solution(s) graphically.
$$\left\{\begin{array}{l}{y=-2 x^{2}+2} \\ {y=2\left(x^{4}-2 x^{2}+1\right)}\end{array}\right.$$

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Problem 21

In Exercises $21-34$ , solve the system by the method of substitution.
\left\{\begin{aligned} x-y &=2 \\ 6 x-5 y &=16 \end{aligned}\right.

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Problem 22

In Exercises $21-34$ , solve the system by the method of substitution.
\left\{\begin{aligned} x+4 y &=3 \\ 2 x-7 y &=-24 \end{aligned}\right.

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Problem 23

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{2 x-y+2=0} \\ {4 x+y-5=0}\end{array}\right.$$

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Problem 24

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{6 x-3 y-4=0} \\ {x+2 y-4=0}\end{array}\right.$$

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Problem 25

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{1.5 x+0.8 y=2.3} \\ {0.3 x-0.2 y=0.1}\end{array}\right.$$

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Problem 26

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{0.5 x+3.2 y=9.0} \\ {0.2 x-1.6 y=-3.6}\end{array}\right.$$

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Problem 27

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{\frac{1}{5} x+\frac{1}{2} y=8} \\ {x+y=20}\end{array}\right.$$

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Problem 28

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{\frac{1}{2} x+\frac{3}{4} y=10} \\ {\frac{3}{4} x-y=4}\end{array}\right.$$

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Problem 29

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{6 x+5 y=-3} \\ {-x-\frac{5}{6} y=-7}\end{array}\right.$$

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Problem 30

In Exercises $21-34$ , solve the system by the method of substitution.
\left\{\begin{aligned}-\frac{2}{3} x+y &=2 \\ 2 x-3 y &=6 \end{aligned}\right.

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Problem 31

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{x^{2}-y=0} \\ {2 x+y=0}\end{array}\right.$$

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Problem 32

In Exercises $21-34$ , solve the system by the method of substitution.
\left\{\begin{aligned} x-2 y &=0 \\ 3 x-y^{2} &=0 \end{aligned}\right.

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Problem 33

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{x-y=-1} \\ {x^{2}-y=-4}\end{array}\right.$$

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Problem 34

In Exercises $21-34$ , solve the system by the method of substitution.
$$\left\{\begin{array}{l}{y=-x} \\ {y=x^{3}+3 x^{2}+2 x}\end{array}\right.$$

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Problem 35

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned}-x+2 y &=-2 \\ 3 x+y &=20 \end{aligned}\right.

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Problem 36

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} x+y &=0 \\ 3 x-2 y &=5 \end{aligned}\right.

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Problem 37

In Exercises $35-48,$ solve the system graphically.
$$\left\{\begin{array}{r}{x-3 y=-3} \\ {5 x+3 y=-6}\end{array}\right.$$

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Problem 38

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned}-x+2 y &=-7 \\ x-y &=2 \end{aligned}\right.

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Problem 39

In Exercises $35-48,$ solve the system graphically.
$$\left\{\begin{array}{r}{x+y=4} \\ {x^{2}+y^{2}-4 x=0}\end{array}\right.$$

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Problem 40

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned}-x+y &=3 \\ x^{2}-6 x-27+y^{2} &=0 \end{aligned}\right.

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Problem 41

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} x-y+3 &=0 \\ x^{2}-4 x+7 &=y \end{aligned}\right.

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Problem 42

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} y^{2}-4 x+11 &=0 \\-\frac{1}{2} x+y &=-\frac{1}{2} \end{aligned}\right.

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Problem 43

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} 7 x+8 y &=24 \\ x-8 y &=8 \end{aligned}\right.

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Problem 44

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} x-y &=0 \\ 5 x-2 y &=6 \end{aligned}\right.

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Problem 45

In Exercises $35-48,$ solve the system graphically.
$$\left\{\begin{array}{l}{3 x-2 y=0} \\ {x^{2}-y^{2}=4}\end{array}\right.$$

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Problem 46

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} 2 x-y+3 &=0 \\ x^{2}+y^{2}-4 x &=0 \end{aligned}\right.

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Problem 47

In Exercises $35-48,$ solve the system graphically.
$$\left\{\begin{array}{c}{x^{2}+y^{2}=25} \\ {3 x^{2}-16 y=0}\end{array}\right.$$

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Problem 48

In Exercises $35-48,$ solve the system graphically.
\left\{\begin{aligned} x^{2}+y^{2} &=25 \\(x-8)^{2}+y^{2} &=41 \end{aligned}\right.

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Problem 49

In Exercises $49-54$ , use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
\left\{\begin{aligned} y &=e^{x} \\ x-y+1 &=0 \end{aligned}\right.

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Problem 50

In Exercises $49-54$ , use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
\left\{\begin{aligned} y &=-4 e^{-x} \\ y+3 x+8 &=0 \end{aligned}\right.

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Problem 51

In Exercises $49-54$ , use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
\left\{\begin{aligned} x+2 y &=8 \\ y &=\log _{2} x \end{aligned}\right.

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Problem 52

In Exercises $49-54$ , use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$$\left\{\begin{array}{c}{y+2=\ln (x-1)} \\ {3 y+2 x=9}\end{array}\right.$$

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Problem 53

In Exercises $49-54$ , use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$$\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x^{2}-8 y=104}\end{array}\right.$$

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Problem 54

In Exercises $49-54$ , use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.
$$\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {2 x^{2}-y=2}\end{array}\right.$$

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Problem 55

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{y=2 x} \\ {y=x^{2}+1}\end{array}\right.$$

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Problem 56

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {2 x+y=10}\end{array}\right.$$

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Problem 57

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{x-2 y=4} \\ {x^{2}-y=0}\end{array}\right.$$

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Problem 58

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{y=(x+1)^{3}} \\ {y=\sqrt{x-1}}\end{array}\right.$$

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Problem 59

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{y-e^{-x}=1} \\ {y-\ln x=3}\end{array}\right.$$

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Problem 60

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{x^{2}+y=4} \\ {e^{x}-y=0}\end{array}\right.$$

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Problem 61

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{y=x^{4}-2 x^{2}+1} \\ {y=1-x^{2}}\end{array}\right.$$

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Problem 62

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
$$\left\{\begin{array}{l}{y=x^{3}-2 x^{2}+x-1} \\ {y=-x^{2}+3 x-1}\end{array}\right.$$

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Problem 63

In Exercises $55-64,$ 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.

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Problem 64

In Exercises $55-64,$ solve the system graphically or algebraically. Explain your choice of method.
\left\{\begin{aligned} x-2 y &=1 \\ y &=\sqrt{x-1} \end{aligned}\right.

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Problem 65

BREAK-EVEN ANALYSIS In Exercises 65 and $66,$ find the sales necessary to break even $(R=C)$ for the cost $C$ of producing $x$ units and the revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
$$C=8650 x+250,000, \quad R=9950 x$$

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Problem 66

BREAK-EVEN ANALYSIS In Exercises 65 and $66,$ find the sales necessary to break even $(R=C)$ for the cost $C$ of producing $x$ units and the revenue $R$ obtained by selling $x$ units. (Round to the nearest whole unit.)
$$C=5.5 \sqrt{x}+10,000, \quad R=3.29 x$$

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Problem 67

BREAK-EVEN ANALYSIS A small software company invests $\$ 25,000$to produce a software package that will sell for$\$69.95 .$ Each unit can be produced for $\$ 45.25$. (a) How many units must be sold to break even? (b) How many units must be sold to make a profit of$\$100,000$ ?

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Problem 68

BREAK-EVEN ANALYSIS A small fast-food restaurant invests $\$ 10,000$to produce a new food item that will sell for$\$3.99 .$ Each item can be produced for $\$ 1.90$. (a) How many items must be sold to break even? (b) How many items must be sold to make a profit of$\$12,000 ?$

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Problem 69

DVD RENTALS The weekly rentals for a newly released DVD of an animated film at a local video store
decreased each week. At the same time, the weekly rentals for a newly released DVD of a horror film increased each week. Models that approximate the weekly rentals $R$ for each DVD are $\left\{\begin{array}{ll}{R=360-24 x} & {\text { Animated film }} \\ {R=24+18 x} & {\text { Horror film }}\end{array}\right.$
where $x$ represents the number of weeks each DVD was in the store, with $x=1$ corresponding to the first week.
(a) After how many weeks will the rentals for the two movies be equal?
(b) Use a table to solve the system of equations numerically. Compare your result with that of part (a).

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Problem 70

SALES The total weekly sales for a newly released portable media player (PMP) increased each week.
At the same time, the total weekly sales for another newly released PMP decreased each week. Models that approximate the total weekly sales $S$ (in thousands of units) for each PMP are $\left\{\begin{array}{ll}{S=} & {15 x+50} & {\text { PMP } 1} \\ {S=} & {-20 x+190} & {\text { PMP } 2}\end{array}\right.$
where $x$ represents the number of weeks each PMP was in stores, with $x=0$ corresponding to the PMP sales on the day each PMP was first released in stores.
(a) After how many weeks will the sales for the two PMPs be equal?
(b) Use a table to solve the system of equations numerically. Compare your result with that of part (a).

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CHOICE OF TWO JOBS You are offered two jobs selling dental supplies. One company offers a straight commission of 6$\%$ of sales. The other company offers a salary of $\$ 500$per week plus 3$\%$of sales. How much would you have to sell in a week in order to make the straight commission offer better? Check back soon! Problem 72 SUPPLY AND DEMAND The supply and demand curves for a business dealing with wheat are Supply:$p=1.45+0.00014 x^{2}$Demand:$p=(2.388-0.007 x)^{2}$where$p$is the price in dollars per bushel and$x$is the quantity in bushels per day. Use a graphing utility to graph the supply and demand equations and find the market equilibrium. (The market equilibrium is the point of intersection of the graphs for$x > 0 .$) Check back soon! Problem 73 INVESTMENT PORTFOLIO A total of$\$25,000$ is invested in two funds paying 6$\%$ and 8.5$\%$ simple interest. (The 6$\%$ investment has a lower risk.) The investor wants a yearly interest income of $\$ 2000$from the two investments. (a) Write a system of equations in which one equation represents the total amount invested and the other equation represents the$\$2000$ required in interest. Let $x$ and $y$ represent the amounts invested at 6$\%$ 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$\%$ increases, how does the amount invested at 8.5$\%$ change? How does the amount of interest income change? Explain.

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