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

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

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

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

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

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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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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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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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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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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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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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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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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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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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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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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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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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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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$$\left\{\begin{array}{l}{y=-2 x^{2}+2} \\ {y=2\left(x^{4}-2 x^{2}+1\right)}\end{array}\right.$$

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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?

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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 $x > 0 .$ )

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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.

(c) What amount should be invested at 6$\%$ to meet the requirement of $\$ 2000$ per year in interest?

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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.) The first rule is the Doyle Log Rule and is modeled by $V_{1}=(D-4)^{2}$ , $5 \leq D \leq 40$ , and the other is the Scribner Log Rule and is modeled by $V_{2}=0.79 D^{2}-2 D-4,5 \leq D \leq 40$ , 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 scales agree?

(c) You are selling large logs by the board foot. Which scale would you use? Explain your reasoning.

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DATA ANALYSIS: RENEWABLE ENERGY The table shows the consumption (in trillions of Btus) of solar

energy and wind energy in the United States from 1998 through 2006.

$$\begin{array}{|c|c|c|}\hline{\text { Year }} & {\text { Solar, } C} & {\text { Wind, } C} \\\hline 1998 & {70} & {31} \\ 1998 & {70} & {31} \\ {2000} & {66} & {57} \\ {2000} & {65} & {70} \\ {2002} & {64} & {105} \\ {2003} & {64} & {115} \\ {2004} & {65} & {142}\\ {2005} & {66} & {178} \\ {2006} & {72} & {264}\\ \hline \end{array}$$

(a) Use the regression feature of a graphing utility to find a cubic model for the solar energy consumption data and a quadratic model for the wind energy consumption data. Let $t$ represent the year, with $t=8$ corresponding to 1998 .

(b) Use a graphing utility to graph the data and the two models in the same viewing window.

(c) Use the graph from part (b) to approximate the point of intersection of the graphs of the models. Interpret your answer in the context of the problem.

(d) Describe the behavior of each model. Do you think the models can be used to predict consumption of solar energy and wind energy in the United States for future years? Explain.

(e) Use your school’s library, the Internet, or some other reference source to research the advantages and disadvantages of using renewable energy.

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DATA ANALYSIS: POPULATION The table shows the populations $P$ (in millions) of Georgia, New Jersey, and North Carolina from 2002 through 2007 . (Source: U.S. Census Bureau)

$$\begin{array}{|r|r|r|r|}\hline \text { Year } & \text { Georgia, }G & \text { New } & \text { North } \\ \text {} & \text {} & \text {Jersey, } J & \text { Carolina, } N \\ \hline {2002} & {8.59} & {8.56} & {8.32} \\ {2003} & {8.74} & {8.61} & {8.42} \\ {2004} & {8.92} & {8.64} & {8.54} \\ {2005} & {9.11} & {8.66} & {8.68} \\ {2006} & {9.34} & {8.67} & {8.87} \\ {2007} & {9.55} & {8.69} & {9.06} \\ \hline \end{array}$$

(a) Use the regression feature of a graphing utility to find linear models for each set of data. Let $t$ represent the year, with $t=2$ corresponding to 2002 .

(b) Use a graphing utility to graph the data and the models in the same viewing window.

(c) Use a graphing utility to graph the data and the models in the same viewing window. points of intersection of the graphs of the models. Interpret the points of intersection in the context of the problem.

(d) Verify your answers from part (c) algebraically.

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DATA ANALYSIS: 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 through 2006 . (Source: U.S. National Center for Education Statistics)

$$\begin{array}{|r|r|}\hline \text { Year } & \text { Public } & \text { Private } \\ \text {} & \text { universities } & \text { universities } \\ \hline {2000} & {2506} & {14,081} \\ {2001} & {2562} & {15,000} \\ {2002} & {2700} & {15,000} \\ {2003} & {2903} & {17,383} \\ {2004} & {3319} & {17,387} \\ {2005} & {3629} & {18,154} \\ {2006} & {3874} & {18,862} \\ \hline \end{array}$$

(a) Use the regression feature of a graphing utility to find a quadratic model $T_{1}$ for tuition at public universities and a linear model $T_{2}$ for tuition at private universities. Let $t$ represent the year, with $t=0$ corresponding to 2000 .

(b) Use a graphing utility to graph the data and the two models in the same viewing window.

(c) Use the graph from part (b) to determine the year after 2006 in which tuition at public universities will exceed tuition at private universities.

(d) Verify your answer from part (c) algebraically.

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GEOMETRY In Exercises 78–82, find the dimensions of the rectangle meeting the specified conditions.

The perimeter is 56 meters and the length is 4 meters greater than the width.

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GEOMETRY In Exercises 78–82, 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.

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GEOMETRY In Exercises 78–82, find the dimensions of the rectangle meeting the specified conditions.

The perimeter is 42 inches and the width is three- fourths the length.

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GEOMETRY In Exercises 78–82, find the dimensions of the rectangle meeting the specified conditions.

The perimeter is 484 feet and the length is times the width.

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GEOMETRY In Exercises 78–82, find the dimensions of the rectangle meeting the specified conditions.

The perimeter is 30.6 millimeters and the length is 2.4 times the width.

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GEOMETRY What are the dimensions of a rectangular tract of land if its perimeter is 44 kilometers and its area is 120 square kilometers?

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

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TRUE OR FALSE? In Exercises 85 and 86, 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 in one of the two equations and then back-substitute.

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TRUE OR FALSE? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer.

If a system consists of a parabola and a circle, then the system can have at most two solutions.

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GRAPHICAL REASONING Use a graphing utility to graph $y_{1}=4-x$ and $y_{2}=x-2$ in the same viewing window. Use the zoom and trace features to find the coordinates of the point of intersection. What is the relationship between the point of intersection and the solution found in Example 1?

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GRAPHICAL REASONING Use a graphing utility to graph the two equations in Example $3,$ $y_{1}=3 x^{2}+4 x-7$ and $y_{2}=2 x+1,$ in the same viewing window. How many solutions do you think this system has? Repeat this experiment for the equations in Example $4 .$ How many solutions does this system have? Explain your reasoning.

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THINK ABOUT IT When solving a system of equations by substitution, how do you recognize that the system has no solution?

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CAPSTONE Consider the system of equations $\left\{\begin{array}{l}{a x+b y=c} \\ {d x+e y=f}\end{array}\right.$

(a) Find values for $a, b, c, d, e,$ and $f$ so that the system has one distinct solution. (There is more than one correct answer.)

(b) Explain how to solve the system in part (a) by the method of substitution and graphically.

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

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Find 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 is more than one correct answer.) Use graphs to support your answers.

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