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Mathematics LibreTexts

9.E: Systems of Equations and Inequalities (Exercises)

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    18811
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    9.1: Systems of Linear Equations: Two Variables

    Verbal

    1) Can a system of linear equations have exactly two solutions? Explain why or why not.

    Answer

    No, you can either have zero, one, or infinitely many. Examine graphs.

    2) If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.

    3) If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?

    Answer

    This means there is no realistic break-even point. By the time the company produces one unit they are already making profit.

    4) If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?

    5) Given a system of equations, explain at least two different methods of solving that system.

    Answer

    You can solve by substitution (isolating \(x\) or \(y\)), graphically, or by addition.

    Algebraic

    For the exercises 6-10, determine whether the given ordered pair is a solution to the system of equations.

    6) \(\begin{align*} 5x-y &= 4\\ x+6y &= 2 \end{align*}\; \text{ and } (4,0)\)

    7) \(\begin{align*} -3x-5y &= 13\\ -x+4y &= 10 \end{align*}\; \text{ and } (-6,1)\)

    Answer

    Yes

    8) \(\begin{align*} 3x+7y &= 1\\ 2x+4y &= 0 \end{align*}\; \text{ and } (2,3)\)

    9) \(\begin{align*} -2x+5y &= 7\\ 2x+9y &= 7 \end{align*}\; \text{ and } (-1,1)\)

    Answer

    Yes

    10) \(\begin{align*} x+8y &= 43\\ 3x-2y &= -1 \end{align*}\; \text{ and } (3,5)\)

    For the exercises 11-20, solve each system by substitution.

    11) \(\begin{align*} x+5y &= 5\\ 2x+3y &= 4 \end{align*}\)

    Answer

    \((-1,2)\)

    12) \(\begin{align*} 3x-2y &= 18\\ 5x+10y &= -10 \end{align*}\)

    13) \(\begin{align*} 4x+2y &= -10\\ 3x+9y &= 0 \end{align*}\)

    Answer

    \((-3,1)\)

    14) \(\begin{align*} 2x+4y &= -3.8\\ 9x-5y &= 1.3 \end{align*}\)

    15) \(\begin{align*} -2x+3y &= 1.2\\ -3x-6y &= 1.8 \end{align*}\)

    Answer

    \(\left ( -\dfrac{3}{5},0 \right )\)

    16) \(\begin{align*} x-0.2y &= 1\\ -10x+2y &= 5 \end{align*}\)

    17) \(\begin{align*} 3x+5y &= 9\\ 30x+50y &= -90 \end{align*}\)

    Answer

    No solutions exist

    18) \(\begin{align*} -3x+y &= 2\\ 12x-4y &= -8 \end{align*}\)

    19) \(\begin{align*} \dfrac{1}{2}x+\dfrac{1}{3}y &= 16\\ \dfrac{1}{6}x+\dfrac{1}{4}y &= 9 \end{align*}\)

    Answer

    \(\left ( \dfrac{72}{5},\dfrac{132}{5} \right )\)

    20) \(\begin{align*} -\dfrac{1}{4}x+\dfrac{3}{2}y &= 11\\ -\dfrac{1}{8}x+\dfrac{1}{3}y &= 3 \end{align*}\)

    For the exercises 21-30, solve each system by addition.

    21) \(\begin{align*} -2x+5y &= -42\\ 7x+2y &= 30 \end{align*}\)

    Answer

    \((6,-6)\)

    22) \(\begin{align*} 6x-5y &= -34\\ 2x+6y &= 4 \end{align*}\)

    23) \(\begin{align*} 5x-y &= -2.6\\ -4x-6y &= 1.4 \end{align*}\)

    Answer

    \(\left ( -\dfrac{1}{2},\dfrac{1}{10} \right )\)

    24) \(\begin{align*} 7x-2y &= 3\\ 4x+5y &= 3.25 \end{align*}\)

    25) \(\begin{align*} -x+2y &= -1\\ 5x-10y &= 6 \end{align*}\)

    Answer

    No solutions exist

    26) \(\begin{align*} 7x+6y &= 2\\ -28x-24y &= -8 \end{align*}\)

    27) \(\begin{align*} \dfrac{5}{6}x+\dfrac{1}{4}y &= 0\\ \dfrac{1}{8}x-\dfrac{1}{2}y &= -\dfrac{43}{120} \end{align*}\)

    Answer

    \(\left ( -\dfrac{1}{5},\dfrac{2}{3} \right )\)

    28) \(\begin{align*} \dfrac{1}{3}x+\dfrac{1}{9}y &= \dfrac{2}{9}\\ -\dfrac{1}{2}x+\dfrac{4}{5}y &= -\dfrac{1}{3} \end{align*}\)

    29) \(\begin{align*} -0.2x+0.4y &= 0.6\\ x-2y &= -3 \end{align*}\)

    Answer

    \(\left ( x,\dfrac{x+3}{2} \right )\)

    30) \(\begin{align*} -0.1x+0.2y &= 0.6\\ 5x-10y &= 1 \end{align*}\)

    For the exercises 31-40, solve each system by any method.

    31) \(\begin{align*} 5x+9y &= 16\\ x+2y &= 4 \end{align*}\)

    Answer

    \((-4,4)\)

    32) \(\begin{align*} 6x-8y &= -0.6\\ 3x+2y &= 0.9 \end{align*}\)

    33) \(\begin{align*} 5x-2y &= 2.25\\ 7x-4y &= 3 \end{align*}\)

    Answer

    \(\left ( \dfrac{1}{2},\dfrac{1}{8} \right )\)

    34) \(\begin{align*} x-\dfrac{5}{12}y &= -\dfrac{55}{12}\\ -6x+\dfrac{5}{2}y &= \dfrac{55}{2} \end{align*}\)

    35) \(\begin{align*} 7x-4y &= \dfrac{7}{6}\\ 2x+4y &= \dfrac{1}{3} \end{align*}\)

    Answer

    \(\left ( \dfrac{1}{6},0 \right )\)

    36) \(\begin{align*} 3x+6y &= 11\\ 2x+4y &= 9 \end{align*}\)

    37) \(\begin{align*} \dfrac{7}{3}x-\dfrac{1}{6}y &= 2\\ -\dfrac{21}{6}x+\dfrac{3}{12}y &= -3 \end{align*}\)

    Answer

    \((x,2(7x-6))\)

    38) \(\begin{align*} \dfrac{1}{2}x+\dfrac{1}{3}y &= \dfrac{1}{3}\\ \dfrac{3}{2}x+\dfrac{1}{4}y &= -\dfrac{1}{8} \end{align*}\)

    39) \(\begin{align*} 2.2x+1.3y &= -0.1\\ 4.2x+4.2y &= 2.1 \end{align*}\)

    Answer

    \(\left ( -\dfrac{5}{6},\dfrac{4}{3} \right )\)

    40) \(\begin{align*} 0.1x+0.2y &= 2\\ 0.35x-0.3y &= 0 \end{align*}\)

    Graphical

    For the exercises 41-45, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

    41) \(\begin{align*} 3x-y &= 0.6\\ x-2y &= 1.3 \end{align*}\)

    Answer

    Consistent with one solution

    42) \(\begin{align*} -x+2y &= 4\\ 2x-4y &= 1 \end{align*}\)

    43) \(\begin{align*} x+2y &= 7\\ 2x+6y &= 12 \end{align*}\)

    Answer

    Consistent with one solution

    44) \(\begin{align*} 3x-5y &= 7\\ x-2y &= 3 \end{align*}\)

    45) \(\begin{align*} 3x-2y &= 5\\ -9x+6y &= -15 \end{align*}\)

    Answer

    Dependent with infinitely many solutions

    Technology

    For the exercises 46-50, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

    46) \(\begin{align*} 0.1x+0.2y &= 0.3\\ -0.3x+0.5y &= 1 \end{align*}\)

    47) \(\begin{align*} -0.01x+0.12y &= 0.62\\ 0.15x+0.20y &= 0.52 \end{align*}\)

    Answer

    \((-3.08,4.91)\)

    48) \(\begin{align*} 0.5x+0.3y &= 4\\ 0.25x-0.9y &= 0.46 \end{align*}\)

    49) \(\begin{align*} 0.15x+0.27y &= 0.39\\ -0.34x+0.56y &= 1.8 \end{align*}\)

    Answer

    \((-1.52,2.29)\)

    50) \(\begin{align*} -0.71x+0.92y &= 0.13\\ 0.83x+0.05y &= 2.1 \end{align*}\)

    Extensions

    For the exercises 51-55, solve each system in terms of \(A, B, C, D,\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A\neq B\) and \(AE\neq BD\).

    51) \(\begin{align*} x+y &= A\\ x-y &= B \end{align*}\)

    Answer

    \(\left ( \dfrac{A+B}{2},\dfrac{A-B}{2} \right )\)

    52) \(\begin{align*} x+Ay &= 1\\ x+By &= 1 \end{align*}\)

    53) \(\begin{align*} Ax+y &= 0\\ Bx+y &= 1 \end{align*}\)

    Answer

    \(\left ( \dfrac{-1}{A-B},\dfrac{A}{A-B} \right )\)

    54) \(\begin{align*} Ax+By &= C\\ x+y &= 1 \end{align*}\)

    55) \(\begin{align*} Ax+By &= C\\ Dx+Ey &= F \end{align*}\)

    Answer

    \(\left ( \dfrac{CE-BF}{BD-AE},\dfrac{AF-CD}{BD-AE} \right )\)

    Real-World Applications

    For the exercises 56-60, solve for the desired quantity.

    56) A stuffed animal business has a total cost of production \(C=12x+30\) and a revenue function \(R=20x\). Find the break-even point.

    57) A fast-food restaurant has a cost of production \(C(x)=11x+120\) and a revenue function \(R(x)=5x\). When does the company start to turn a profit?

    Answer

    They never turn a profit.

    58) A cell phone factory has a cost of productiona \(C(x)=150x+10,000\) and a revenue function \(R(x)=200x\). What is the break-even point?

    59) A musician charges \(C(x)=64x+20,000\), where \(x\) is the total number of attendees at the concert. The venue charges \(\$80\) per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

    Answer

    \((1,250, 100,000)\)

    60) A guitar factory has a cost of production \(C(x)=75x+50,000\). If the company needs to break even after \(150\) units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

    For the exercises 61-77, use a system of linear equations with two variables and two equations to solve.

    61) Find two numbers whose sum is \(28\) and difference is \(13\).

    Answer

    The numbers are \(7.5\) and \(20.5\)

    62) A number is \(9\) more than another number. Twice the sum of the two numbers is \(10\). Find the two numbers.

    63) The startup cost for a restaurant is \(\$120,000\), and each meal costs \(\$10\) for the restaurant to make. If each meal is then sold for \(\$15\), after how many meals does the restaurant break even?

    Answer

    \(24,000\)

    64) A moving company charges a flat rate of \(\$150\), and an additional \(\$5\) for each box. If a taxi service would charge \(\$20\) for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

    65) A total of \(1,595\) first- and second-year college students gathered at a pep rally. The number of freshmen exceeded the number of sophomores by \(15\). How many freshmen and sophomores were in attendance?

    Answer

    \(790\) sophomores, \(805\) freshman

    66) \(276\) students enrolled in a freshman-level chemistry class. By the end of the semester, \(5\) times the number of students passed as failed. Find the number of students who passed, and the number of students who failed.

    67) There were \(130\) faculty at a conference. If there were \(18\) more women than men attending, how many of each gender attended the conference?

    Answer

    \(56\) men, \(74\) women

    68) A jeep and BMW enter a highway running east-west at the same exit heading in opposite directions. The jeep entered the highway \(30\) minutes before the BMW did, and traveled \(7\) mph slower than the BMW. After \(2\) hours from the time the BMW entered the highway, the cars were \(306.5\) miles apart. Find the speed of each car, assuming they were driven on cruise control.

    69) If a scientist mixed \(10\%\) saline solution with \(60\%\) saline solution to get \(25\) gallons of \(40\%\) saline solution, how many gallons of \(10\%\) and \(60\%\) solutions were mixed?

    Answer

    \(10\) gallons of \(10\%\) solution, \(15\) gallons of \(60\%\) solution

    70) An investor earned triple the profits of what she earned last year. If she made \(\$500,000.48\) total for both years, how much did she earn in profits each year?

    71) An investor who dabbles in real estate invested \(1.1\) million dollars into two land investments. On the first investment, Swan Peak, her return was a \(110\%\) increase on the money she invested. On the second investment, Riverside Community, she earned \(50\%\) over what she invested. If she earned \(\$1\) million in profits, how much did she invest in each of the land deals?

    Answer

    Swan Peak: \(\$750,000\), Riverside: \(\$350,000\)

    72) If an investor invests a total of \(\$25,000\) into two bonds, one that pays \(3\%\) simple interest, and the other that pays \(2\dfrac{7}{8}\%\) interest, and the investor earns \(\$737.50\) annual interest, how much was invested in each account?

    73) If an investor invests \(\$23,000\) into two bonds, one that pays \(4\%\) in simple interest, and the other paying \(2\%\) simple interest, and the investor earns \(\$710.00\) annual interest, how much was invested in each account?

    Answer

    \(\$12,500\) in the first account, \(\$10,500\) in the second account.

    74) CDs cost \(\$5.96\) more than DVDs at All Bets Are Off Electronics. How much would \(6\) CDs and \(2\) DVDs cost if \(5\) CDs and \(2\) DVDs cost \(\$127.73\)?

    75) A store clerk sold \(60\) pairs of sneakers. The high-tops sold for \(\$98.99\) and the low-tops sold for \(\$129.99\). If the receipts for the two types of sales totaled \(\$6,404.40\), how many of each type of sneaker were sold?

    Answer

    High-tops: \(45\), Low-tops: \(15\)

    76) A concert manager counted \(350\) ticket receipts the day after a concert. The price for a student ticket was \(\$12.50\), and the price for an adult ticket was \(\$16.00\). The register confirms that \(\$5,075\) was taken in. How many student tickets and adult tickets were sold?

    77) Admission into an amusement park for \(4\) children and \(2\) adults is \(\$116.90\). For \(6\) children and \(3\) adults, the admission is \(\$175.35\). Assuming a different price for children and adults, what is the price of the child’s ticket and the price of the adult ticket?

    Answer

    Infinitely many solutions. We need more information.

    9.2: Systems of Linear Equations: Three Variables

    Verbal

    1) Can a linear system of three equations have exactly two solutions? Explain why or why not

    Answer

    No, there can be only one, zero, or infinitely many solutions.

    2) If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.

    3) If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.

    Answer

    Not necessarily. There could be zero, one, or infinitely many solutions. For example, \((0,0,0)\) is not a solution to the system below, but that does not mean that it has no solution.

    \(\begin{align*} 2x+3y-6z &= 1\\ -4x-6y+12z &= -2\\ x+2y+5z &= 10 \end{align*}\)

    4) Using the method of addition, is there only one way to solve the system?

    5) Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

    Answer

    Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.

    Algebraic

    For the exercises 6-10, determine whether the ordered triple given is the solution to the system of equations.

    6) \(\begin{align*} 2x-6y+6z &= -12\\ x+4y+5z &= -1\\ -x+2y+3z &= -1 \end{align*}\; \; \text{ and }\; (0,1,-1)\)

    7) \(\begin{align*} 6x-y+3z &= 6\\ 3x+5y+2z &= 0\\ x+y &= 0 \end{align*}\; \; \text{ and }\; (3,-3,-5)\)

    Answer

    No

    8) \(\begin{align*} 6x-7y+z &= 2\\ -x-y+3z &= 4\\ 2x+y-z &= 1 \end{align*}\; \; \text{ and }\; (4,2,-6)\)

    9) \(\begin{align*} x-y &= 0\\ x-z &= 5\\ x-y+z &= -1 \end{align*}\; \; \text{ and }\; (4,4,-1)\)

    Answer

    Yes

    10) \(\begin{align*} -x-y+2z &= 3\\ 5x+8y-3z &= 4\\ -x+3y-5z &= -5 \end{align*}\; \; \text{ and }\; (4,1,-7)\)

    For the exercises 11-16, solve each system by substitution.

    11) \(\begin{align*} 3x-4y+2z &= -15\\ 2x+4y+z &= 16\\ 2x+3y+5z &= 20 \end{align*}\)

    Answer

    \((-1,4,2)\)

    12) \(\begin{align*} 5x-2y+3z &= 20\\ 2x-4y-3z &= -9\\ x+6y-8z &= 21 \end{align*}\)

    13) \(\begin{align*} 5x+2y+4z &= 9\\ -3x+2y+z &= 10\\ 4x-3y+5z &= -3 \end{align*}\)

    Answer

    \(\left ( -\dfrac{85}{107},\dfrac{312}{107},\dfrac{191}{107} \right )\)

    14) \(\begin{align*} 4x-3y+5z &= 31\\ -x+2y+4z &= 20\\ x+5y-2z &= -29 \end{align*}\)

    15) \(\begin{align*} 5x-2y+3z &= 4\\ -4x+6y-7z &= -1\\ 3x+2y-z &= 4 \end{align*}\)

    Answer

    \(\left ( 1,\dfrac{1}{2},0 \right )\)

    16) \(\begin{align*} 4x+6y+9z &= 4\\ -5x+2y-6z &= 3\\ 7x-4y+3z &= -3 \end{align*}\)

    For the exercises 17-45, solve each system by Gaussian elimination.

    17) \(\begin{align*} 2x-y+3z &= 17\\ -5x+4y-2z &= -46\\ 2y+5z &= -7 \end{align*}\)

    Answer

    \((4,-6,1)\)

    18) \(\begin{align*} 5x-6y+3z &= 50\\ -x+4y &= 10\\ 2x-z &= 10 \end{align*}\)

    19) \(\begin{align*} 2x+3y-6z &= 1\\ -4x-6y+12z &= -2\\ x+2y+5z &= 10 \end{align*}\)

    Answer

    \(\left ( x,\dfrac{1}{27}(65-16x),\dfrac{x+28}{27} \right )\)

    20) \(\begin{align*} 4x+6y-2z &= 8\\ 6x+9y-3z &= 12\\ -2x-3y+z &= -4 \end{align*}\)

    21) \(\begin{align*} 2x+3y-4z &= 5\\ -3x+2y+z &= 11\\ -x+5y+3z &= 4 \end{align*}\)

    Answer

    \(\left ( -\dfrac{45}{13},\dfrac{17}{13},-2 \right )\)

    22) \(\begin{align*} 10x+2y-14z &= 8\\ -x-2y-4z &= -1\\ -12x-6y+6z &= -12 \end{align*}\)

    23) \(\begin{align*} x+y+z &= 14\\ 2y+3z &= -14\\ -16y-24z &= -112 \end{align*}\)

    Answer

    No solutions exist

    24) \(\begin{align*} 5x-3y+4z &= -1\\ -4x+2y-3z &= 0\\ -x+5y+7z &= -11 \end{align*}\)

    25) \(\begin{align*} x+y+z &= 0\\ 2x-y+3z &= 0\\ x-z &= 0 \end{align*}\)

    Answer

    \((0,0,0)\)

    26) \(\begin{align*} 3x+2y-5z &= 6\\ 5x-4y+3z &= -12\\ 4x+5y-2z &= 15 \end{align*}\)

    27) \(\begin{align*} x+y+z &= 0\\ 2x-y+3z &= 0\\ x-z &= 1 \end{align*}\)

    Answer

    \(\left ( \dfrac{4}{7},-\dfrac{1}{7},-\dfrac{3}{7} \right )\)

    28) \(\begin{align*} 3x-\dfrac{1}{2}y-z &= -\dfrac{1}{2}\\ 4x+z &= 3\\ -x+\dfrac{3}{2}y &= \dfrac{5}{2} \end{align*}\)

    29) \(\begin{align*} 6x-5y+6z &= 38\\ \dfrac{1}{5}x-\dfrac{1}{2}y+\dfrac{3}{5}z &= 1\\ -4x-\dfrac{3}{2}y-z &= -74 \end{align*}\)

    Answer

    \((7,20,16)\)

    30) \(\begin{align*} \dfrac{1}{2}x-\dfrac{1}{5}y+\dfrac{2}{5}z &= -\dfrac{13}{10}\\ \dfrac{1}{4}x-\dfrac{2}{5}y-\dfrac{1}{5}z &= -\dfrac{7}{20}\\ -\dfrac{1}{2}x-\dfrac{3}{4}y-\dfrac{1}{2}z &= -\dfrac{5}{4} \end{align*}\)

    31) \(\begin{align*} -\dfrac{1}{3}x-\dfrac{1}{2}y-\dfrac{1}{4}z &= \dfrac{3}{4}\\ -\dfrac{1}{2}x-\dfrac{1}{4}y-\dfrac{1}{2}z &= 2\\ -\dfrac{1}{4}x-\dfrac{3}{4}y-\dfrac{1}{2}z &= -\dfrac{1}{2} \end{align*}\)

    Answer

    \((-6,2,1)\)

    32) \(\begin{align*} \dfrac{1}{2}x-\dfrac{1}{4}y+\dfrac{3}{4}z &= 0\\ \dfrac{1}{4}x-\dfrac{1}{10}y+\dfrac{2}{5}z &= -2\\ \dfrac{1}{8}x+\dfrac{1}{5}y-\dfrac{1}{8}z &= 2 \end{align*}\)

    33) \(\begin{align*} \dfrac{4}{5}x-\dfrac{7}{8}y+\dfrac{1}{2}z &= 1\\ -\dfrac{4}{5}x-\dfrac{3}{4}y+\dfrac{1}{3}z &= -8\\ -\dfrac{2}{5}x-\dfrac{7}{8}y+\dfrac{1}{2}z &= -5 \end{align*}\)

    Answer

    \((5,12,15)\)

    34) \(\begin{align*} -\dfrac{1}{3}x-\dfrac{1}{8}y+\dfrac{1}{6}z &= -\dfrac{4}{3}\\ -\dfrac{2}{3}x-\dfrac{7}{8}y+\dfrac{1}{3}z &= -\dfrac{23}{3}\\ -\dfrac{1}{3}x-\dfrac{5}{8}y+\dfrac{5}{6}z &= 0 \end{align*}\)

    35) \(\begin{align*} -\dfrac{1}{4}x-\dfrac{5}{4}y+\dfrac{5}{2}z &= -5\\ -\dfrac{1}{2}x-\dfrac{5}{3}y+\dfrac{5}{4}z &= \dfrac{55}{12}\\ -\dfrac{1}{3}x-\dfrac{1}{3}y+\dfrac{1}{3}z &= \dfrac{5}{3} \end{align*}\)

    Answer

    \((-5,-5,-5)\)

    36) \(\begin{align*} \dfrac{1}{40}x+\dfrac{1}{60}y+\dfrac{1}{80}z &= \dfrac{1}{100}\\ -\dfrac{1}{2}x-\dfrac{1}{3}y-\dfrac{1}{4}z &= -\dfrac{1}{5}\\ \dfrac{3}{8}x+\dfrac{3}{12}y+\dfrac{3}{16}z &= \dfrac{3}{20} \end{align*}\)

    37) \(\begin{align*} 0.1x-0.2y+0.3z &= 2\\ 0.5x-0.1y+0.4z &= 8\\ 0.7x-0.2y+0.3z &= 8 \end{align*}\)

    Answer

    \((10,10,10)\)

    38) \(\begin{align*} 0.2x+0.1y-0.3z &= 0.2\\ 0.8x+0.4y-1.2z &= 0.1\\ 1.6x+0.8y-2.4z &= 0.2 \end{align*}\)

    39) \(\begin{align*} 1.1x+0.7y-3.1z &= -1.79\\ 2.1x+0.5y-1.6z &= -0.13\\ 0.5x+0.4y-0.5z &= -0.07 \end{align*}\)

    Answer

    \(\left ( \dfrac{1}{2},\dfrac{1}{5},\dfrac{4}{5} \right )\)

    40) \(\begin{align*} 0.5x-0.5y+0.5z &= 10\\ 0.2x-0.2y+0.2z &= 4\\ 0.1x-0.1y+0.1z &= 2 \end{align*}\)

    41) \(\begin{align*} 0.1x+0.2y+0.3z &= 0.37\\ 0.1x-0.2y-0.3z &= -0.27\\ 0.5x-0.1y-0.3z &= -0.03 \end{align*}\)

    Answer

    \(\left ( \dfrac{1}{2},\dfrac{2}{5},\dfrac{4}{5} \right )\)

    42) \(\begin{align*} 0.5x-0.5y-0.3z &= 0.13\\ 0.4x-0.1y-0.3z &= 0.11\\ 0.2x-0.8y-0.9z &= -0.32 \end{align*}\)

    43) \(\begin{align*} 0.5x+0.2y-0.3z &= 1\\ 0.4x-0.6y+0.7z &= 0.8\\ 0.3x-0.1y-0.9z &= 0.6 \end{align*}\)

    Answer

    \((2,0,0)\)

    44) \(\begin{align*} 0.3x+0.3y+0.5z &= 0.6\\ 0.4x+0.4y+0.4z &= 1.8\\ 0.4x+0.2y+0.1z &= 1.6 \end{align*}\)

    45) \(\begin{align*} 0.8x+0.8y+0.8z &= 2.4\\ 0.3x-0.5y+0.2z &= 0\\ 0.1x+0.2y+0.3z &= 0.6 \end{align*}\)

    Answer

    \((1,1,1)\)

    Extensions

    For the exercises 46-50, solve the system for \(x,y,\) and \(z\).

    46) \(\begin{align*} x+y+z &= 3\\ \dfrac{x-1}{2}+\dfrac{y-3}{2}+\dfrac{z+1}{2} &= 0\\ \dfrac{x-2}{3}+\dfrac{y+4}{3}+\dfrac{z-3}{3} &= \dfrac{2}{3} \end{align*}\)

    47) \(\begin{align*} 5x-3y-\dfrac{z+1}{2} &= \dfrac{1}{2}\\ 6x+\dfrac{y-9}{2}+2z &= -3\\ \dfrac{x+8}{2}-4y+z &= 4\end{align*}\)

    Answer

    \(\left ( \dfrac{128}{557},\dfrac{23}{557},\dfrac{428}{557} \right )\)

    48) \(\begin{align*} \dfrac{x+4}{7}-\dfrac{y-1}{6}+\dfrac{z+2}{3} &= 1\\ \dfrac{x-2}{4}+\dfrac{y+1}{8}-\dfrac{z+8}{2} &= 0\\ \dfrac{x+6}{3}-\dfrac{y+2}{3}+\dfrac{z+4}{2} &= 3 \end{align*}\)

    49) \(\begin{align*} \dfrac{x-3}{6}+\dfrac{y+2}{2}-\dfrac{z-3}{3} &= 2\\ \dfrac{x+2}{4}+\dfrac{y-5}{2}+\dfrac{z+4}{2} &= 1\\ \dfrac{x+6}{2}-\dfrac{y-3}{3}+z+1 &= 9 \end{align*}\)

    Answer

    \((6,-1,0)\)

    50) \(\begin{align*} \dfrac{x-1}{3}+\dfrac{y+3}{4}+\dfrac{z+2}{6} &= 1\\ 4x+3y-2z &= 11\\ 0.02x+0.015y-0.01z &= 0.065 \end{align*}\)

    Real-World Applications

    51) Three even numbers sum up to \(108\). The smaller is half the larger and the middle number is \(\dfrac{3}{4}\) the larger. What are the three numbers?

    Answer

    \(24, 36, 48\)

    52) Three numbers sum up to \(147\). The smallest number is half the middle number, which is half the largest number. What are the three numbers?

    53) At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were \(400\) people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?

    Answer

    \(70\) grandparents, \(140\) parents, \(190\) children

    54) An animal shelter has a total of \(350\) animals comprised of cats, dogs, and rabbits. If the number of rabbits is \(5\) less than one-half the number of cats, and there are \(20\) more cats than dogs, how many of each animal are at the shelter?

    55) Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was \(\$82\). She forgot to save the individual receipts but remembered that your groceries were \(\$0.05\) cheaper than half of her groceries, and that your other roommate’s groceries were \(\$2.10\) more than your groceries. How much was each of your share of the groceries?

    Answer

    Your share was \(\$19.95\), Sarah’s share was \(\$40\), and your other roommate’s share was \(\$22.05\).

    56) Your roommate, John, offered to buy household supplies for you and your other roommate. You live near the border of three states, each of which has a different sales tax. The total amount of money spent was \(\$100.75\). Your supplies were bought with \(5\%\) tax, John’s with \(8\%\) tax, and your third roommate’s with \(9\%\) sales tax. The total amount of money spent without taxes is \(\$93.50\). If your supplies before tax were \(\$1\) more than half of what your third roommate’s supplies were before tax, how much did each of you spend? Give your answer both with and without taxes.

    57) Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is \(\$82,000\). The office manager makes \(\$4,000\) more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total \(\$78,000\). What is the annual salary of each of the co-workers?

    Answer

    There are infinitely many solutions; we need more information

    58) At a carnival, \(\$2,914.25\) in receipts were taken at the end of the day. The cost of a child’s ticket was \(\$20.50\), an adult ticket was \(\$29.75\), and a senior citizen ticket was \(\$15.25\). There were twice as many senior citizens as adults in attendance, and \(20\) more children than senior citizens. How many children, adult, and senior citizen tickets were sold?

    59) A local band sells out for their concert. They sell all \(1,175\) tickets for a total purse of \(\$28,112.50\). The tickets were priced at \(\$20\) for student tickets, \(\$22.50\) for children, and \(\$29\) for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

    Answer

    \(500\) students, \(225\) children, and \(450\) adults

    60) In a bag, a child has \(325\) coins worth \(\$19.50\). There were three types of coins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?

    61) Last year, at Haven’s Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of \(\$140,000\). This year, due to inflation, the same cars would cost \(\$151,830\). The cost of the BMW increased by \(8\%\), the Jeep by \(5\%\), and the Toyota by \(12\%\). If the price of last year’s Jeep was \(\$7,000\) less than the price of last year’s BMW, what was the price of each of the three cars last year?

    Answer

    The BMW was \(\$49,636\), the Jeep was \(\$42,636\), and the Toyota was \(\$47,727\).

    62) A recent college graduate took advantage of his business education and invested in three investments immediately after graduating. He invested \(\$80,500\) into three accounts, one that paid \(4\%\) simple interest, one that paid \(3\dfrac{1}{8}\%\) simple interest, and one that paid \(2\dfrac{1}{2}\%\) simple interest. He earned \(\$2,670\) interest at the end of one year. If the amount of the money invested in the second account was four times the amount invested in the third account, how much was invested in each account?

    63) You inherit one million dollars. You invest it all in three accounts for one year. The first account pays \(3\%\) compounded annually, the second account pays \(4\%\) compounded annually, and the third account pays \(2\%\) compounded annually. After one year, you earn \(\$34,000\) in interest. If you invest four times the money into the account that pays \(3\%\) compared to \(2\%\), how much did you invest in each account?

    Answer

    \(\$400,000\) in the account that pays \(3\%\) interest, \(\$500,000\) in the account that pays \(4\%\) interest, and \(\$100,000\) in the account that pays \(2\%\) interest.

    64) You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays \(4\%\) compounded annually, the second account pays \(3\%\) compounded annually, and the third account pays \(2\%\) compounded annually. After one year, you earn \(\$3,650\) in interest. If you invest five times the money in the account that pays \(4\%\) compared to \(3\%\), how much did you invest in each account?

    65) The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed \(39.8\%\) of the world’s consumed oil. The United States consumed \(0.7\%\) more than four times China’s consumption. The United States consumed \(5\%\) more than triple Japan’s consumption. What percent of the world oil consumption did the United States, Japan, and China consume?

    Answer

    The United States consumed \(26.3\%\), Japan \(7.1\%\), and China \(6.4\%\) of the world’s oil.

    66) The top three countries in oil production in the same year are Saudi Arabia, the United States, and Russia. In millions of barrels per day, the top three countries produced \(31.4\%\) of the world’s produced oil. Saudi Arabia and the United States combined for \(22.1\%\) of the world’s production, and Saudi Arabia produced \(2\%\) more oil than Russia. What percent of the world oil production did Saudi Arabia, the United States, and Russia produce?

    67) The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for \(47\%\) of oil imports. The United States imported \(1.8\%\) more from Saudi Arabia than they did from Mexico, and \(1.7\%\) more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?

    Answer

    Saudi Arabia imported \(16.8\%\), Canada imported \(15.1\%\), and Mexico \(15.0\%\)

    68) The top three oil producers in the United States in a certain year are the Gulf of Mexico, Texas, and Alaska. The three regions were responsible for \(64\%\) of the United States oil production. The Gulf of Mexico and Texas combined for \(47\%\) of oil production. Texas produced \(3\%\) more than Alaska. What percent of United States oil production came from these regions?

    69) At one time, in the United States, \(398\) species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised \(55\%\) of the endangered species. Birds accounted for \(0.7\%\) more than fish, and fish accounted for \(1.5\%\) more than mammals. What percent of the endangered species came from mammals, birds, and fish?

    Answer

    Birds were \(19.3\%\), fish were \(18.6\%\), and mammals were \(17.1\%\) of endangered species

    70) Meat consumption in the United States can be broken into three categories: red meat, poultry, and fish. If fish makes up \(4\%\) less than one-quarter of poultry consumption, and red meat consumption is \(18.2\%\) higher than poultry consumption, what are the percentages of meat consumption?

    9.3: Systems of Nonlinear Equations and Inequalities: Two Variables

    Verbal

    1) Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three? If not, explain why not. If so, give an example of such a system, in graph form, and explain why your choice gives two or three answers.

    Answer

    A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.

    2) When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?

    3) When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?

    Answer

    No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

    4) If you graph a revenue and cost function, explain how to determine in what regions there is profit.

    5) If you perform your break-even analysis and there is more than one solution, explain how you would determine which x-values are profit and which are not.

    Answer

    Choose any number between each solution and plug into \(C(x)\) and \(R(x)\). If \(C(x)<R(x)\), then there is profit.

    Algebraic

    For the exercises 6-10, solve the system of nonlinear equations using substitution.

    6) \(\begin{align*} x+y &= 4\\ x^2 + y^2 &= 9 \end{align*}\)

    7) \(\begin{align*} y &= x-3\\ x^2 + y^2 &= 9 \end{align*}\)

    Answer

    \((0,-3)\), \((3,0)\)

    8) \(\begin{align*} y &= x\\ x^2 + y^2 &= 9 \end{align*}\)

    9) \(\begin{align*} y &= -x\\ x^2 + y^2 &= 9 \end{align*}\)

    Answer

    \(\left ( -\dfrac{3\sqrt{2}}{2},\dfrac{3\sqrt{2}}{2} \right )\), \(\left ( \dfrac{3\sqrt{2}}{2},-\dfrac{3\sqrt{2}}{2} \right )\)

    10) \(\begin{align*} x &= 2\\ x^2 - y^2 &= 9 \end{align*}\)

    For the exercises 11-15, solve the system of nonlinear equations using elimination.

    11) \(\begin{align*} 4x^2 - 9y^2 &= 36\\ 4x^2 + 9y^2 &= 36 \end{align*}\)

    Answer

    \((-3,0)\), \((3,0)\)

    12) \(\begin{align*} x^2 + y^2 &= 25\\ x^2 - y^2 &= 1 \end{align*}\)

    13) \(\begin{align*} 2x^2 + 4y^2 &= 4\\ 2x^2 - 4y^2 &= 25x-10 \end{align*}\)

    Answer

    \(\left ( \dfrac{1}{4},-\dfrac{\sqrt{62}}{8} \right )\), \(\left ( \dfrac{1}{4},\dfrac{\sqrt{62}}{8} \right )\)

    14) \(\begin{align*} y^2 - x^2 &= 9\\ 3x^2 + 2y^2 &= 8 \end{align*}\)

    15) \(\begin{align*} x^2 + y^2+\dfrac{1}{16} &= 2500\\ y &= 2x^2 \end{align*}\)

    Answer

    \(\left ( -\dfrac{\sqrt{398}}{4},\dfrac{199}{4} \right )\), \(\left ( \dfrac{\sqrt{398}}{4},\dfrac{199}{4} \right )\)

    For the exercises 16-23, use any method to solve the system of nonlinear equations.

    16) \(\begin{align*} -2x^2+y &= -5\\ 6x-y &= 9 \end{align*}\)

    17) \(\begin{align*} -x^2+y &= 2\\ -x+y &= 2 \end{align*}\)

    Answer

    \((0,2)\), \((1,3)\)

    18) \(\begin{align*} x^2+y^2 &= 1\\ y &= 20x^2-1 \end{align*}\)

    19) \(\begin{align*} x^2+y^2 &= 1\\ y &= -x^2 \end{align*}\)

    Answer

    \(\left ( -\sqrt{\dfrac{1}{2}(\sqrt{5}-1)},\dfrac{1}{2}\left (1-\sqrt{5} \right ) \right )\), \(\left ( \sqrt{\dfrac{1}{2}(\sqrt{5}-1)},\dfrac{1}{2}\left (1-\sqrt{5} \right ) \right )\)

    20) \(\begin{align*} 2x^3-x^2 &= y\\ y &= \dfrac{1}{2} -x \end{align*}\)

    21) \(\begin{align*} 9x^2+25y^2 &= 225\\ (x-6)^2+y^2 &= 1 \end{align*}\)

    Answer

    \((5,0)\)

    22) \(\begin{align*} x^4-x^2 &= y\\ x^2+y &= 0 \end{align*}\)

    23) \(\begin{align*} 2x^3-x^2 &= y\\ x^2+y &= 0 \end{align*}\)

    Answer

    \((0,0)\)

    For the exercises 24-38, use any method to solve the nonlinear system.

    24) \(\begin{align*} x^2+y^2 &= 9\\ y &= 3-x^2 \end{align*}\)

    25) \(\begin{align*} x^2-y^2 &= 9\\ x &= 3 \end{align*}\)

    Answer

    \((3,0)\)

    26) \(\begin{align*} x^2-y^2 &= 9\\ y &= 3 \end{align*}\)

    27) \(\begin{align*} x^2-y^2 &= 9\\ x-y &= 0 \end{align*}\)

    Answer

    No Solutions Exist

    28) \(\begin{align*} -x^2+y &= 2\\ -4x+y &= -1 \end{align*}\)

    29) \(\begin{align*} -x^2+y &= 2\\ 2y &= -x \end{align*}\)

    Answer

    No Solutions Exist

    30) \(\begin{align*} x^2+y^2 &= 25\\ x^2-y^2 &= 36 \end{align*}\)

    31) \(\begin{align*} x^2+y^2 &= 1\\ y^2 &= x^2 \end{align*}\)

    Answer

    \(\left ( -\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2} \right )\), \(\left ( -\dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right )\), \(\left ( \dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2} \right )\), \(\left ( \dfrac{\sqrt{2}}{2},\dfrac{\sqrt{2}}{2} \right )\)

    32) \(\begin{align*} 16x^2-9y^2+144 &= 0\\ y^2 + x^2 &= 16 \end{align*}\)

    33) \(\begin{align*} 3x^2-y^2 &= 12\\ (x-1)^2 + y^2 &= 1 \end{align*}\)

    Answer

    \((2,0)\)

    34) \(\begin{align*} 3x^2-y^2 &= 12\\ (x-1)^2 + y^2 &= 4 \end{align*}\)

    35) \(\begin{align*} 3x^2-y^2 &= 12\\ x^2 + y^2 &= 16 \end{align*}\)

    Answer

    \((-\sqrt{7},-3)\), \((-\sqrt{7},3)\), \((\sqrt{7},-3)\), \((\sqrt{7},3)\)

    36) \(\begin{align*} x^2-y^2-6x-4y-11 &= 0\\ -x^2 + y^2 &= 5 \end{align*}\)

    37) \(\begin{align*} x^2+y^2-6y &= 7\\ x^2 + y &= 1 \end{align*}\)

    Answer

    \(\left ( -\sqrt{\dfrac{1}{2}(\sqrt{73}-5)},\dfrac{1}{2}\left (7-\sqrt{73} \right ) \right )\), \(\left ( \sqrt{\dfrac{1}{2}(\sqrt{73}-5)},\dfrac{1}{2}\left (7-\sqrt{73} \right ) \right )\)

    38) \(\begin{align*} x^2+y^2 &= 6\\ xy &= 1 \end{align*}\)

    Graphical

    For the exercises 39-40, graph the inequality.

    39) \(x^2+y<9\)

    Answer

    CNX_Precalc_Figure_09_03_201.jpg

    40) \(x^2+y^2<4\)

    For the exercises 41-45, graph the system of inequalities. Label all points of intersection.

    41) \(\begin{align*} x^2 + y &<1 \\ y &>2x \end{align*}\)

    Answer

    CNX_Precalc_Figure_09_03_204.jpg

    42) \(\begin{align*} x^2 + y &<-5 \\ y &>5x+10 \end{align*}\)

    43) \(\begin{align*} x^2 + y^2 &<25 \\ 3x^2 - y^2 &>12 \end{align*}\)

    Answer

    CNX_Precalc_Figure_09_03_207.jpg

    44) \(\begin{align*} x^2 - y^2 &>-4 \\ x^2 + y^2 &<12 \end{align*}\)

    45) \(\begin{align*} x^2 + 3y^2 &>16 \\ 3x^2 - y^2 &<1 \end{align*}\)

    Answer

    CNX_Precalc_Figure_09_03_209.jpg

    Extensions

    For the exercises 46-47, graph the inequality.

    46) \(\begin{align*} y &\geq e^x \\ y &\leq \ln (x)+5 \end{align*}\)

    47) \(\begin{align*} y &\leq -\log (x)\\ y &\leq e^x \end{align*}\)

    Answer

    CNX_Precalc_Figure_09_03_210.jpg

    For the exercises 48-52, find the solutions to the nonlinear equations with two variables.

    48) \(\begin{align*} \dfrac{4}{x^2} + \dfrac{1}{y^2} &= 24\\ \dfrac{5}{x^2} - \dfrac{2}{y^2} + 4 &= 0 \end{align*}\)

    49) \(\begin{align*} \dfrac{6}{x^2} - \dfrac{1}{y^2} &= 8\\ \dfrac{1}{x^2} - \dfrac{6}{y^2} &= \dfrac{1}{8} \end{align*}\)

    Answer

    \(\left ( -2\sqrt{\dfrac{70}{383}},-2\sqrt{\dfrac{35}{29}} \right )\), \(\left ( -2\sqrt{\dfrac{70}{383}},2\sqrt{\dfrac{35}{29}} \right )\), \(\left ( 2\sqrt{\dfrac{70}{383}},-2\sqrt{\dfrac{35}{29}} \right )\), \(\left ( 2\sqrt{\dfrac{70}{383}},2\sqrt{\dfrac{35}{29}} \right )\)

    50) \(\begin{align*} x^2 - xy + y^2 - 2 &= 0\\ x+3y &= 4 \end{align*}\)

    51) \(\begin{align*} x^2 - xy - 2y^2 - 6 &= 0\\ x^2 + y^2 &= 1 \end{align*}\)

    Answer

    No Solution Exists

    52) \(\begin{align*} x^2 + 4xy - 2y^2 - 6 &= 0\\ x &= y+2 \end{align*}\)

    Technology

    For the exercises 53-54, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

    53) \(\begin{align*} xy &< 1\\ y &> \sqrt{x} \end{align*}\)

    Answer

    \(x=0\), \(y>0\) and \(0<x<1\), \(\sqrt{x} < y < \dfrac{1}{x}\)

    54) \(\begin{align*} x^2 + y &< 3\\ y &> 2x \end{align*}\)

    Real-World Applications

    For the exercises 55-, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.

    55) Two numbers add up to \(300\). One number is twice the square of the other number. What are the numbers?

    Answer

    \(12,288\)

    56) The squares of two numbers add to \(360\). The second number is half the value of the first number squared. What are the numbers?

    57) A laptop company has discovered their cost and revenue functions for each day: \(C(x)=3x^2-10x+200\) and \(R(x)=-2x^2+100x+50\). If they want to make a profit, what is the range of laptops per day that they should produce? Round to the nearest number which would generate profit.

    Answer

    \(2\) - \(20\) computers

    58) A cell phone company has the following cost and revenue functions: \(C(x)=8x^2-600x+21,500\) and \(R(x)=-3x^2+480x\). What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

    9.4: Partial Fractions

    Verbal

    1) Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction.

    Answer

    No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, \(\dfrac{1}{x^2+1}\) cannot be decomposed because the denominator cannot be factored.

    2) Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)

    3) Can you explain how to verify a partial fraction decomposition graphically?

    Answer

    Graph both sides and ensure they are equal.

    4) You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.

    5) Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had \(\dfrac{7x+13}{3x^2+8x+15}=\dfrac{A}{x+1}+\dfrac{B}{3x+5}\) we eventually simplify to \(7x+13=A(3x+5)+B(x+1)\). Explain how you could intelligently choose an \(x\)-value that will eliminate either \(A\) or \(B\) and solve for \(A\) and \(B\).

    Answer

    If we choose \(x=-1\),then the \(B\)-term disappears, letting us immediately know that \(A=3\). We could alternatively plug in \(x=-\dfrac{5}{3}\),giving us a \(B\)-value of