🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Precalculus with Limits ## Ron Larson ## Chapter 7 ## Systems of Equations and Inequalities ## Educators DD CC ### Problem 1 Fill in the blanks. A set of two or more equations in two or more variables is called a ________ of ________. DD Daniel D. Community College of the Air Force ### Problem 2 Fill in the blanks. A ________ of a system of equations is an ordered pair that satisfies each equation in the system. DD Daniel D. Community College of the Air Force ### Problem 3 Fill in the blanks. Finding the set of all solutions to a system of equations is called ________ the system of equations. DD Daniel D. Community College of the Air Force ### Problem 4 Fill in the blanks. The first step in solving a system of equations by the method of ________ is to solve one of the equations for one variable in terms of the other variable. DD Daniel D. Community College of the Air Force ### Problem 5 Fill in the blanks. Graphically, the solution of a system of two equations is the ________ of ________ of the graphs of the two equations. Julie S. Numerade Educator ### Problem 6 Fill in the blanks. In business applications, the point at which the revenue equals costs is called the ________ point. Julie S. Numerade Educator ### Problem 7 In Exercises 7 - 10, determine whether each ordered pair is a solution of the system of equations$ \left\{\begin{array}{l} 2x - y = 4\\8x + y = -9\end{array}\right. $(a)$ (0 , -4) $(b)$ (-2 , 7) $(c)$ (\dfrac{3}{2} , -1) $(d)$ (- \dfrac{1}{2} , -5) $CC Charles C. Numerade Educator ### Problem 8 In Exercises 7 - 10, determine whether each ordered pair is a solution of the system of equations$ \left\{\begin{array}{l}4x^2 + y = 3\\-x - y = 11\end{array}\right. $(a)$ (2 , -13) $(b)$ (2 , -9) $(c)$ (- \dfrac{3}{2} , - \dfrac{31}{3}) $(d)$ (- \dfrac{7}{4} , - \dfrac{37}{4}) $Alisa L. Numerade Educator ### Problem 9 In Exercises 7 - 10, determine whether each ordered pair is a solution of the system of equations$ \left\{\begin{array}{l} \hspace{1cm} y = -4e^x\\7x - y = 4\end{array}\right. $(a)$ (-4 , 0) $(b)$ (0 , -4) $(c)$ (0 , -2) $(d)$ (-1 , -3) $CC Charles C. Numerade Educator ### Problem 10 In Exercises 7 - 10, determine whether each ordered pair is a solution of the system of equations$ \left\{\begin{array}{l}- \log x + 3 = y\\-\dfrac{1}{9}x + y = \dfrac{28}{9}\end{array}\right. $(a)$ (9 , \dfrac{37}{9}) $(b)$ (10 , 2) $(c)$ (1 , 3) $(d)$ (2 , 4) $Alisa L. Numerade Educator ### Problem 11 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}2x + y = 6\\-x + y = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 12 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}x - 4y = -11\\x + 3y = 3\end{array}\right. $Alisa L. Numerade Educator ### Problem 13 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}x - y = -4\\x^2 - y = -2\end{array}\right. $CC Charles C. Numerade Educator ### Problem 14 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l} \hspace{1cm} 3x + y = 2\\x^3 - 2 + y = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 15 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}-\dfrac{1}{2}x + y = -\dfrac{5}{2}\\x^2 + y^2 = 25\end{array}\right. $CC Charles C. Numerade Educator ### Problem 16 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l} \hspace{1cm} x + y = 0\\x^3 - 5x - y = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 17 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l} \hspace{1cm} x^2 + t = 0\\x^2 - 4x - y = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 18 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}y = -2x^2 + 2\\y = 2\left(x^4 - 2x^2 +1\right)\end{array}\right. $Alisa L. Numerade Educator ### Problem 19 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}y = x^3 - 3x^2 + 1\\y = x^2 - 3x + 1\end{array}\right. $CC Charles C. Numerade Educator ### Problem 20 In Exercises 11 - 20, solve the system by the method of substitution. Check your solution(s) graphically.$ \left\{\begin{array}{l}y = x^3 - 3x^2 + 4\\y = -2x + 4\end{array}\right. $Alisa L. Numerade Educator ### Problem 21 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}x - y = 2\\6x - 5y = 16\end{array}\right. $CC Charles C. Numerade Educator ### Problem 22 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}x + 4y = 3\\2x - 7y = -24\end{array}\right. $Alisa L. Numerade Educator ### Problem 23 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}2x - y + 2 = 0\\4x + y - 5 = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 24 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}6x - 3y - 4 = 0\\x + 2y - 4 = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 25 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}1.5x + 0.8y = 2.3\\0.3x - 0.2y = 0.1\end{array}\right. $CC Charles C. Numerade Educator ### Problem 26 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}0.5x + 3.2y = 9.0\\0.2x - 1.6y = -3.6\end{array}\right. $Alisa L. Numerade Educator ### Problem 27 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}\dfrac{1}{5}x + \dfrac{1}{2}y = 8\\x + y = 20\end{array}\right. $CC Charles C. Numerade Educator ### Problem 28 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}\dfrac{1}{2}x + \dfrac{3}{4}y = 10\\\dfrac{3}{4}x - y = 4\end{array}\right. $Alisa L. Numerade Educator ### Problem 29 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}6x + 5y = -3\\-x - \dfrac{5}{6}y = -7\end{array}\right. $CC Charles C. Numerade Educator ### Problem 30 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}- \dfrac{2}{3}x + y = 2\\2x - 3y = 6\end{array}\right. $Alisa L. Numerade Educator ### Problem 31 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}x^2 - y = 0\\2x + y = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 32 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}x - 2y = 0\\3x - y^2 = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 33 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}x - y = -1\\x^2 - y = -4\end{array}\right. $CC Charles C. Numerade Educator ### Problem 34 In Exercises 21 - 34, solve the system by the method of substitution.$ \left\{\begin{array}{l}y = -x\\y = x^3 + 3x^2 + 2x\end{array}\right. $Alisa L. Numerade Educator ### Problem 35 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}-x + 2y = -2\\3x + y = 20\end{array}\right. $CC Charles C. Numerade Educator ### Problem 36 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}x + y = 0\\3x - 2y = 5\end{array}\right. $Alisa L. Numerade Educator ### Problem 37 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}x - 3y = -3\\5x + 3y = -6\end{array}\right. $CC Charles C. Numerade Educator ### Problem 38 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}-x + 2y = -7\\x - y = 2\end{array}\right. $Alisa L. Numerade Educator ### Problem 39 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l} \hspace{1cm} x + y = 4\\x^2 + y^2 - 4x = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 40 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l} \hspace{1cm} \hspace{1cm} -x + y = 3\\x^2 - 6x - 27 + y^2 = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 41 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}x - y + 3 = 0\\x^2 - 4x + 7 = y\end{array}\right. $CC Charles C. Numerade Educator ### Problem 42 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l} y^2 - 4x + 11 = 0\\ \hspace{1cm} -\dfrac{1}{2}x + y = -\dfrac{1}{2}\end{array}\right. $Alisa L. Numerade Educator ### Problem 43 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}7x + 8y = 24\\x - 8y = 8\end{array}\right. $CC Charles C. Numerade Educator ### Problem 44 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}x - y = 0\\5x - 2y = 6\end{array}\right. $Alisa L. Numerade Educator ### Problem 45 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}3x - 2y = 0\\x^2 - y^2 = 4\end{array}\right. $CC Charles C. Numerade Educator ### Problem 46 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}2x - y + 3 = 0\\x^2 + y^2 - 4x = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 47 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l}x^2 + y^2 = 25\\3x^2 - 16y = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 48 In Exercises 35 - 48, solve the system graphically.$ \left\{\begin{array}{l} \hspace{1cm} x^2 + y^2 = 25\\\left(x - 8\right)^2 + y^2 = 41\end{array}\right. $Alisa L. Numerade Educator ### Problem 49 In Exercises 49 - 54, use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.$ \left\{\begin{array}{l} \hspace{1cm} \hspace{1cm} y = e^x\\x - y + 1 = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 50 In Exercises 49 - 54, use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.$ \left\{\begin{array}{l} \hspace{1cm} \hspace{1cm} y = -4e^{-x}\\y + 3x + 8 = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 51 In Exercises 49 - 54, use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.$ \left\{\begin{array}{l}x + 2y = g\\ \hspace{1cm} y = \log_2 x\end{array}\right. $CC Charles C. Numerade Educator ### Problem 52 In Exercises 49 - 54, use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.$ \left\{\begin{array}{l}y + 2 = \ln\left(x - 1\right)\\3y + 2x = 9\end{array}\right. $Alisa L. Numerade Educator ### Problem 53 In Exercises 49 - 54, use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.$ \left\{\begin{array}{l}x^2 + y^2 = 169\\x^2 - 8y = 104\end{array}\right. $CC Charles C. Numerade Educator ### Problem 54 In Exercises 49 - 54, use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places.$ \left\{\begin{array}{l}x^2 + y^2 = 4\\2x^2 - y = 2\end{array}\right. $Alisa L. Numerade Educator ### Problem 55 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}y = 2x\\y = x^2 + 1\end{array}\right. $CC Charles C. Numerade Educator ### Problem 56 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}x^2 + y^2 = 25\\2x + y = 10\end{array}\right. $Alisa L. Numerade Educator ### Problem 57 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}x - 2y = 4\\x^2 - y = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 58 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}y = \left(x + 1\right)^3\\y = \sqrt{x - 1}\end{array}\right. $Alisa L. Numerade Educator ### Problem 59 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}y - e^{-x} = 1\\y - \ln x = 3\end{array}\right. $CC Charles C. Numerade Educator ### Problem 60 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}x^2 + y = 4\\e^x - y = 0\end{array}\right. $Alisa L. Numerade Educator ### Problem 61 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}y = x^4 - 2x^2 + 1\\y = 1 - x^2\end{array}\right. $CC Charles C. Numerade Educator ### Problem 62 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}y = x^3 - 2x^2 + x - 1\\y = -x^2 + 3x - 1\end{array}\right. $Alisa L. Numerade Educator ### Problem 63 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l} \hspace{1cm} xy - 1 = 0\\2x - 4y + 7 = 0\end{array}\right. $CC Charles C. Numerade Educator ### Problem 64 In Exercises 55 - 64, solve the system graphically or algebraically. Explain your choice of method.$ \left\{\begin{array}{l}x - 2y = 1\\ \hspace{1cm} y = \sqrt{x - 1}\end{array}\right. $Alisa L. Numerade Educator ### Problem 65 In Exercises 65 and 66, find the sales necessary to break even$ \left(R = C\right) $for the cost$ C $of producing$ x $units and the revenue$ R $obtained by selling$ x $units. (Round to the nearest whole unit.)$ C = 8650x + 250,000 $,$ R = 9950x $CC Charles C. Numerade Educator ### Problem 66 In Exercises 65 and 66, find the sales necessary to break even$ \left(R = C\right) $for the cost$ 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 $,$ R = 3.29 $Alisa L. Numerade Educator ### Problem 67 NO QUESTION YET! Noah M. Numerade Educator ### Problem 68 A small fast-food restaurant invests$ \$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?

CC
Charles C.

### Problem 72

The supply and demand curves for a business dealing with wheat are

Supply: $p = 1.45 + 0.00014x^2$

Demand: $p = \left(2.388 - 0.007x\right)^2$

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 $x > 0$. )

Alisa L.

### Problem 73

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 $required in interest.Let and represent the amounts invested at$ 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\% $to meet therequirement of$ \$2000$ per year in interest?

CC
Charles C.

### Problem 74

You are offered two different rules for estimating the number of board feet in a $16$-foot log. (Aboard foot is a unit of measure for lumber equal to aboard $1$ foot square and $1$ inch thick.) The first rule is the Doyle Log Rule and is modeled by $V_1 = \left(D - 4\right)^2$, $5 \le D \le 40$, and the other is the Scribner Log Rule and is modeled by $V_2 = 0.79D^2 - 2D - 4$, $5 \le D \le 40$, 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.

Alisa L.

### Problem 75

The table shows the consumption $C$ (in trillions of Btus) of solar energy and wind energy in the United States from $1998$ through $2006$.(Source: Energy Information Administration)

(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 schools library, the Internet, or some other reference source to research the advantages and disadvantages of using renewable energy.

Noah M.

### Problem 76

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

(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 the graph from part (b) to approximate any points of intersection of the graphs of the models.Interpret the points of intersection in the context of the problem.

Noah M.

### Problem 77

The table shows the average costs (in dollars) of one years tuition for public and private universities in the United States from $2000$ through $2006$.(Source: U.S. National Center for Education Statistics)

(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 $2000$.

Noah M.

### Problem 78

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.

Alisa L.