Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
[ "article:topic", "authorname:openstax", "license:ccby", "showtoc:yes" ]
Mathematics LibreTexts

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

  • Page ID
    13543
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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

    Algebraic

    For the exercises 6-19, find the decomposition of the partial fraction for the nonrepeating linear factors.

    6) \(\dfrac{5x+16}{x^2+10x+24}\)

    7) \(\dfrac{3x-79}{x^2-5x-24}\)

    Answer

    \(\dfrac{8}{x+3}-\dfrac{5}{x-8}\)

    8) \(\dfrac{-x-24}{x^2-2x-24}\)

    9) \(\dfrac{10x+47}{x^2+7x+10}\)

    Answer

    \(\dfrac{1}{x+5}+\dfrac{9}{x+2}\)

    10) \(\dfrac{x}{6x^2+25x+25}\)

    11) \(\dfrac{32x-11}{20x^2-13x+2}\)

    Answer

    \(\dfrac{3}{5x-2}+\dfrac{4}{4x-1}\)

    12) \(\dfrac{x+1}{x^2+7x+10}\)

    13) \(\dfrac{5x}{x^2-9}\)

    Answer

    \(\dfrac{5}{2(x+3)}+\dfrac{5}{2(x-3)}\)

    14) \(\dfrac{10x}{x^2-25}\)

    15) \(\dfrac{6x}{x^2-4}\)

    Answer

    \(\dfrac{3}{x+2}+\dfrac{3}{x-2}\)

    16) \(\dfrac{2x-3}{x^2-6x+5}\)

    17) \(\dfrac{4x-1}{x^2-x-6}\)

    Answer

    \(\dfrac{9}{5(x+2)}+\dfrac{11}{5(x-3)}\)

    18) \(\dfrac{4x+3}{x^2+8x+15}\)

    19) \(\dfrac{3x-1}{x^2-5x+6}\)

    Answer

    \(\dfrac{8}{x-3}-\dfrac{5}{x-2}\)

    For the exercises 20-30, find the decomposition of the partial fraction for the repeating linear factors.

    20) \(\dfrac{-5x-19}{(x+4)^2}\)

    21) \(\dfrac{x}{(x-2)^2}\)

    Answer

    \(\dfrac{1}{x-2}-\dfrac{2}{(x-2)^2}\)

    22) \(\dfrac{7x+14}{(x+3)^2}\)

    23) \(\dfrac{-24x-27}{(4x+5)^2}\)

    Answer

    \(-\dfrac{6}{4x+5}+\dfrac{3}{(4x+5)^2}\)

    24) \(\dfrac{-24x-27}{(6x-7)^2}\)

    25) \(\dfrac{5-x}{(x-7)^2}\)

    Answer

    \(-\dfrac{1}{x-7}-\dfrac{2}{(x-7)^2}\)

    26) \(\dfrac{5x+14}{2x^2+12x+18}\)

    27) \(\dfrac{5x^2+20x+8}{2x(x+1)^2}\)

    Answer

    \(\dfrac{4}{x}-\dfrac{3}{2(x+1)}+\dfrac{7}{2(x+1)^2}\)

    28) \(\dfrac{4x^2+55x+25}{5x(3x+5)^2}\)

    29) \(\dfrac{54x^3+127x^2+80x+16}{2x^2(3x+2)^2}\)

    Answer

    \(\dfrac{4}{x}+\dfrac{2}{x^2}-\dfrac{3}{3x+2}+\dfrac{7}{2(3x+2)^2}\)

    30) \(\dfrac{x^3-5x^2+12x+144}{x^2(x^2+12x+36)}\)

    For the exercises 31-43, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

    31) \(\dfrac{4x^2+6x+11}{(x+2)(x^2+x+3)}\)

    Answer

    \(\dfrac{x+1}{x^2+x+3}+\dfrac{3}{(x+2)}\)

    32) \(\dfrac{4x^2+9x+23}{(x-1)(x^2+6x+11)}\)

    33) \(\dfrac{-2x^2+10x+4}{(x-1)(x^2+3x+8)}\)

    Answer

    \(\dfrac{4-3x}{x^2+3x+8}+\dfrac{1}{(x-1)}\)

    34) \(\dfrac{x^2+3x+1}{(x+1)(x^2+5x-2)}\)

    35) \(\dfrac{4x^2+17x-1}{(x+3)(x^2+6x+1)}\)

    Answer

    \(\dfrac{2x-1}{x^2+6x+1}+\dfrac{2}{(x+3)}\)

    36) \(\dfrac{4x^2}{(x+5)(x^2+7x-5)}\)

    37) \(\dfrac{4x^2+x+3}{x^3 - 1}\)

    Answer

    \(\dfrac{1}{x^2+x+1}+\dfrac{4}{(x-1)}\)

    38) \(\dfrac{-5x^2+18x-4}{x^3 + 8}\)

    39) \(\dfrac{3x^2-7x+33}{x^3 + 27}\)

    Answer

    \(\dfrac{2}{x^2-3x+9}+\dfrac{3}{(x+3)}\)

    40) \(\dfrac{x^2+2x+40}{x^3 - 125}\)

    41) \(\dfrac{4x^2+4x+12}{8x^3 - 27}\)

    Answer

    \(-\dfrac{1}{4x^2+6x+9}+\dfrac{1}{(2x-3)}\)

    42) \(\dfrac{-50x^2+5x-3}{125x^3 - 1}\)

    43) \(\dfrac{-2x^3-30x^2+36x+216}{x^4 + 216x}\)

    Answer

    \(\dfrac{1}{x}+\dfrac{1}{x+6}-\dfrac{4x}{x^2-6x+36}\)

    For the exercises 44-54, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

    44) \(\dfrac{3x^3+2x^2+14x+15}{(x^2 + 4)^2}\)

    45) \(\dfrac{x^3+6x^2+5x+9}{(x^2 + 1)^2}\)

    Answer

    \(\dfrac{x+6}{x^2+1}+\dfrac{4x+3}{(x^2+1)^2}\)

    46) \(\dfrac{x^3-x^2+x-1}{(x^2 - 3)^2}\)

    47) \(\dfrac{x^2+5x+5}{(x+2)^2}\)

    Answer

    \(\dfrac{x+1}{x+2}+\dfrac{2x+3}{(x+2)^2}\)

    48) \(\dfrac{x^3+2x^2+4x}{(x^2+2x+9)^2}\)

    49) \(\dfrac{x^2+25}{(x^2+3x+25)^2}\)

    Answer

    \(\dfrac{1}{x^2+3x+25}-\dfrac{3x}{(x^2+3x+25)^2}\)

    50) \(\dfrac{2x^3+11x+7x+70}{(2x^2+x+14)^2}\)

    51) \(\dfrac{5x+2}{x(x^2+4)^2}\)

    Answer

    \(\dfrac{1}{8x}-\dfrac{x}{8(x^2+4)}+\dfrac{10-x}{8(x^2+4)^2}\)

    52) \(\dfrac{x^4+x^3+8x^2+6x+36}{x(x^2+6)^2}\)

    53) \(\dfrac{2x-9}{(x^2-x)^2}\)

    Answer

    \(-\dfrac{16}{x}-\dfrac{9}{x^2}+\dfrac{16}{x-1}-\dfrac{7}{(x-1)^2}\)

    54) \(\dfrac{5x^3-2x+1}{(x^2+2x)^2}\)

    Extensions

    For the exercises 55-56, find the partial fraction expansion.

    55) \(\dfrac{x^2+4}{(x+1)^3}\)

    Answer

    \(\dfrac{1}{x+1}-\dfrac{2}{(x+1)^2}+\dfrac{5}{(x+1)^3}\)

    56) \(\dfrac{x^3-4x^2+5x+4}{(x-2)^3}\)

    For the exercises 57-59, perform the operation and then find the partial fraction decomposition.

    57) \(\dfrac{7}{x+8}+\dfrac{5}{x-2}-\dfrac{x-1}{x^2-6x-16}\)

    Answer

    \(\dfrac{5}{x-2}-\dfrac{3}{10(x+2)}+\dfrac{7}{x+8}-\dfrac{7}{10(x-8)}\)

    58) \(\dfrac{1}{x-4}-\dfrac{3}{x+6}-\dfrac{2x+7}{x^2+2x-24}\)

    59) \(\dfrac{2x}{x^2-16}-\dfrac{1-2x}{x^2+6x+8}-\dfrac{x-5}{x^2-4x}\)

    Answer

    \(-\dfrac{5}{4x}-\dfrac{5}{2(x+2)}+\dfrac{11}{2(x+4)}+\dfrac{5}{4(x+4)}\)

    9.5: Matrices and Matrix Operations

    Verbal

    1) Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

    Answer

    No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a \(2\times 2\) matrix and the second is a

    2) Can we multiply any column matrix by any row matrix? Explain why or why not.

    3) Can both the products \(AB\) and \(BA\) be defined? If so, explain how; if not, explain why.

    Answer

    Yes, if the dimensions of \(A\) are \(m\times n\) and the dimensions of \(B\) are \(n\times m\) both products will be defined.

    4) Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

    5) Does matrix multiplication commute? That is, does \(AB=BA\)? If so, prove why it does. If not, explain why it does not.

    Answer

    Not necessarily. To find \(AB\), we multiply the first row of \(A\) by the first column of \(B\) to get the first entry of \(AB\). To find \(BA\), we multiply the first row of \(B\) by the first column of \(A\) to get the first entry of \(BA\). Thus, if those are unequal, then the matrix multiplication does not commute.

    Algebraic

    For the exercises 6-11, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

    \[A=\begin{bmatrix} 1 & 3\\ 0 & 7 \end{bmatrix}, B=\begin{bmatrix} 2 & 14\\ 22 & 6 \end{bmatrix}, C=\begin{bmatrix} 1 & 5\\ 8 & 92\\ 12 & 6 \end{bmatrix}, D=\begin{bmatrix} 10 & 14\\ 7 & 2\\ 5 & 61 \end{bmatrix}, E=\begin{bmatrix} 6 & 12\\ 14 & 5 \end{bmatrix}, F=\begin{bmatrix} 0 & 9\\ 78 & 17\\ 15 & 4 \end{bmatrix} \nonumber\]

    6) \(A+B\)

    7) \(C+D\)

    Answer

    \(\begin{bmatrix} 11 & 19\\ 15 & 94\\ 17 & 67 \end{bmatrix}\)

    8) \(A+C\)

    9) \(B-E\)

    Answer

    \(\begin{bmatrix} -4 & 2\\ 8 & 1 \end{bmatrix}\)

    10) \(C+F\)

    11) \(D-B\)

    Answer

    Undefined; dimensions do not match

    For the exercises 12-17, use the matrices below to perform scalar multiplication.

    \[A=\begin{bmatrix} 4 & 6\\ 13 & 12 \end{bmatrix}, B=\begin{bmatrix} 3 & 9\\ 21 & 12\\ 0 & 64 \end{bmatrix}, C=\begin{bmatrix} 16 & 3 & 7 & 18\\ 90 & 5 & 3 & 29 \end{bmatrix}, D=\begin{bmatrix} 18 & 12 & 13\\ 8 & 14 & 6\\ 7 & 4 & 21 \end{bmatrix} \nonumber\]

    12) \(5A\)

    13) \(3B\)

    Answer

    \(\begin{bmatrix} 9 & 27\\ 63 & 36\\ 0 & 192 \end{bmatrix}\)

    14) \(-2B\)

    15) \(-4C\) 

    Answer

    \(\begin{bmatrix} -64 & -12 & -28 & -72\\ -360 & -20 & -12 & -116 \end{bmatrix}\)

    16) \(\dfrac{1}{2}C\)

    17) \(100D\)

    Answer

    \(\begin{bmatrix} 1,800 & 1,200 & 1,300\\ 800 & 1,400 & 600\\ 700 & 400 & 2,100 \end{bmatrix}\)

    For the exercises 18-23, use the matrices below to perform matrix multiplication.

    \[A=\begin{bmatrix} -1 & 5\\ 3 & 2 \end{bmatrix}, B=\begin{bmatrix} 3 & 6 & 4\\ -8 & 0 & 12 \end{bmatrix}, C=\begin{bmatrix} 4 & 10\\ -2 & 6\\ 5 & 9 \end{bmatrix}, D=\begin{bmatrix} 2 & -3 & 12\\ 9 & 3 & 1\\ 0 & 8 & -10 \end{bmatrix} \nonumber\]

    18) \(AB\)

    19) \(BC\)

    Answer

    \(\begin{bmatrix} 20 & 102\\ 28 & 28 \end{bmatrix}\)

    20) \(CA\)

    21) \(BD\)

    Answer

    \(\begin{bmatrix} 60 & 41 & 2\\ -16 & 120 & -216 \end{bmatrix}\)

    22) \(DC\)

    23) \(CB\)

    Answer

    \(\begin{bmatrix} -68 & 24 & 136\\ -54 & -12 & 64\\ -57 & 30 & 128 \end{bmatrix}\)

    For the exercises 24-29, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

    \[A=\begin{bmatrix} 2 & -5\\ 6 & 7 \end{bmatrix}, B=\begin{bmatrix} -9 & 6\\ -4 & 2 \end{bmatrix}, C=\begin{bmatrix} 0 & 9\\ 7 & 1 \end{bmatrix}, D=\begin{bmatrix} -8 & 7 & -5\\ 4 & 3 & 2\\ 0 & 9 & 2 \end{bmatrix}, E=\begin{bmatrix} 4 & 5 & 3\\ 7 & -6 & -5\\ 1 & 0 & 9 \end{bmatrix} \nonumber\]

    24) \(A+B-C\)

    25) \(4A+5D\)

    Answer

    Undefined; dimensions do not match.

    26) \(2C+B\)

    27) \(3D+4E\)

    Answer

    \(\begin{bmatrix} -8 & 41 & -3\\ 40 & -15 & -14\\ 4 & 27 & 42 \end{bmatrix}\)

    28) \(C-0.5D\)

    29) \(100D-10E\)

    Answer

    \(\begin{bmatrix} -840 & 650 & -530\\ 330 & 360 & 250\\ -10 & 900 & 110 \end{bmatrix}\)

    For the exercises 30-40, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^2=A\cdot A\))

    \[A=\begin{bmatrix} -10 & 20\\ 5 & 25 \end{bmatrix}, B=\begin{bmatrix} 40 & 10\\ -20 & 30 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\\ 0 & -1\\ 1 & 0 \end{bmatrix} \nonumber\]

    30) \(AB\)

    31) \(BA\)

    Answer

    \(\begin{bmatrix} -350 & 1,050\\ 350 & 350 \end{bmatrix}\)

    32) \(CA\)

    33) \(BC\)

    Answer

    Undefined; inner dimensions do not match.

    34) \(A^2\)

    35) \(B^2\)

    Answer

    \(\begin{bmatrix} 1,400 & 700\\ -1,400 & 700 \end{bmatrix}\)

    36) \(C^2\)

    37) \(B^2A^2\)

    Answer

    \(\begin{bmatrix} 332,500 & 927,500\\ -227,500 & 87,500 \end{bmatrix}\)

    38) \(A^2B^2\)

    39) \((AB)^2\)

    Answer

    \(\begin{bmatrix} 490,000 & 0\\ 0 & 490,000 \end{bmatrix}\)

    40)\((BA)^2\)

    For the exercises 41-49, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^2=A\cdot A\))

    \[A=\begin{bmatrix} 1 & 0\\ 2 & 3 \end{bmatrix}, B=\begin{bmatrix} -2 & 3 & 4\\ -1 & 1 & -5 \end{bmatrix}, C=\begin{bmatrix} 0.5 & 0.1\\ 1 & 0.2\\ -0.5 & 0.3 \end{bmatrix}, D=\begin{bmatrix} 1 & 0 & -1\\ -6 & 7 & 5\\ 4 & 2 & 1 \end{bmatrix} \nonumber\]

    41) \(AB\)

    Answer

    \(\begin{bmatrix} -2 & 3 & 4\\ -7 & 9 & -7 \end{bmatrix} \nonumber\)

    42) \(BA\)

    43) \(BD\)

    Answer

    \(\begin{bmatrix} -4 & 29 & 21\\ -27 & -3 & 1 \end{bmatrix} \nonumber\)

    44) \(DC\)

    45) \(D^2\)

    Answer

    \(\begin{bmatrix} -3 & -2 & -2\\ -28 & 59 & 46\\ -4 & 16 & 7 \end{bmatrix} \nonumber\)

    46) \(A^2\)

    47) \(D^3\)

    Answer

    \(\begin{bmatrix} 1 & -18 & -9\\ -198 & 505 & 369\\ -72 & 126 & 91 \end{bmatrix} \nonumber\)

    48) \((AB)C\)

    49) \(A(BC)\)

    Answer

    \(\begin{bmatrix} 0 & 1.6\\ 9 & -1 \end{bmatrix} \nonumber\)

    Technology

    For the exercises 50-54, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

    \[A=\begin{bmatrix} -2 & 0 & 9\\ 1 & 8 & -3\\ 0.5 & 4 & 5 \end{bmatrix}, B=\begin{bmatrix} 0.5 & 3 & 0\\ -4 & 1 & 6\\ 8 & 7 & 2 \end{bmatrix}, C=\begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} \nonumber\]

    50) \(AB\)

    51) \(BA\)

    Answer

    \(\begin{bmatrix} 2 & 24 & -4.5\\ 12 & 32 & -9\\ -8 & 64 & 61 \end{bmatrix} \nonumber\)

    52) \(CA\)

    53) \(BC\)

    Answer

    \(\begin{bmatrix} 0.5 & 3 & 0.5\\ 2 & 1 & 2\\ 10 & 7 & 10 \end{bmatrix} \nonumber\)

    54) \(ABC\)

    Extensions

    For the exercises 55-, use the matrix below to perform the indicated operation on the given matrix.

    \[B=\begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix} \nonumber\]

    55) \(B^2\)

    Answer

    \(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \nonumber\)

    56) \(B^3\)

    57) \(B^4\)

    Answer

    \(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \nonumber\)

    58) \(B^5\)

    59) Using the above questions, find a formula for \(B^n\). Test the formula for \(B^{201}\) and \(B^{202}\), using a calculator.

    Answer

    \(B^n=\begin{cases} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}, & n\text{ even }\\ \\ \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix}, & n\text{ odd } \end{cases}\)

    9.6: Solving Systems with Gaussian Elimination

    Verbal

    1) Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

    Answer

    Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.

    2) Can any matrix be written as a system of linear equations? Explain why or why not. Explain how to write that system of equations.

    3) Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix \(\left [ \begin{array}{cc|c} 9 & 3 & 0\\ 1 & -2 & 6\\ \end{array} \right ]\).

    Answer

    No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following:

    1. Interchange rows 1 and 2. Then \(R_2=R_2-9R_1\).
    2. \(R_2=R_1-9R_2\) .   Then divide row 1 by \(9\).

    4) Can a matrix whose entry is \(0\) on the diagonal be solved? Explain why or why not. What would you do to remedy the situation?

    5) Can a matrix that has \(0\) entries for an entire row have one solution? Explain why or why not.

    Answer

    No. A matrix with \(0\) entries for an entire row would have either zero or infinitely many solutions.

    Algebraic

    For the exercises 6-10, write the augmented matrix for the linear system.

    6) \(\begin{align*} 8x-37y &= 8\\ 2x+12y &= 3 \end{align*}\)

    7) \(\begin{align*} 16y &= 4\\ 9x-y &= 2 \end{align*}\)

    Answer

    \(\left [ \begin{array}{cc|c} 0 & 16 & 4\\ 9 & -1 & 2\\ \end{array} \right ]\)

    8) \(\begin{align*} 3x+2y+10z &= 3\\ -6x+2y+5z &= 13\\ 4x+z &= 18 \end{align*}\)

    9) \(\begin{align*} x+5y+8z &= 19\\ 12x+3y &= 4\\ 3x+4y+9z &= -7 \end{align*}\)

    Answer

    \(\left [ \begin{array}{ccc|c} 1 & 5 & 8 & 16\\ 12 & 3 & 0 & 4\\ 3 & 4 & 9 & -7\end{array} \right ]\)

    10) \(\begin{align*} 6x+12y+16z &= 4\\ 19x-5y+3z &= -9\\ x+2y &= -8 \end{align*}\)

    For the exercises 11-15, write the linear system from the augmented matrix.

    11) \(\left [ \begin{array}{cc|c} -2 & 5 & 5\\ 6 & -18 & 26\\ \end{array} \right ]\)

    Answer

    \(\begin{align*} -2x+5y &= 5\\ 6x-18y &= 26 \end{align*}\)

    12) \(\left [ \begin{array}{cc|c} 3 & 4 & 10\\ 10 & 17 & 439\\ \end{array} \right ]\)

    13) \(\left [ \begin{array}{ccc|c} 3 & 2 & 0 & 3\\ -1 & -9 & 4& -1\\ 8 & 5 & 7 & 8\\ \end{array} \right ]\)

    Answer

    \(\begin{align*} 3x+2y &= 13\\ -x-9y+4z &= 53\\ 8x+5y+7z &= 80 \end{align*}\)

    14) \(\left [ \begin{array}{ccc|c} 8 & 29 & 1 & 43\\ -1 & 7 & 5 & 38\\ 0 & 0 & 3 & 10\\ \end{array} \right ]\)

    15) \(\left [ \begin{array}{ccc|c} 4 & 5 & -2 & 12\\ 0 & 1 & 58 & 2\\ 8 & 7 & -3 & -5\\ \end{array} \right ]\)

    Answer

    \(\begin{align*} 4x+5y-2z &= 12\\ y+58z &= 2\\ 8x+7y-3z &= -5 \end{align*}\)

    For the exercises 16-46, solve the system by Gaussian elimination.

    16) \(\left [ \begin{array}{cc|c} 1 & 0 & 3\\ 0 & 0 & 0\\ \end{array} \right ]\)

    17) \(\left [ \begin{array}{cc|c} 1 & 0 & 1\\ 1 & 0 & 2\\ \end{array} \right ]\)

    Answer

    No solutions

    18) \(\left [ \begin{array}{cc|c} 1 & 2 & 3\\ 4 & 5 & 6\\ \end{array} \right ]\)

    19) \(\left [ \begin{array}{cc|c} -1 & 2 & -3\\ 4 & -5 & 6\\ \end{array} \right ]\)

    Answer

    \((-1,-2)\)

    20) \(\left [ \begin{array}{cc|c} -2 & 0 & 1\\ 0 & 2 & -1\\ \end{array} \right ]\)

    21) \(\begin{align*} 2x-3y &= -9\\ 5x+4y &= 58 \end{align*}\)

    Answer

    \((6,7)\)

    22) \(\begin{align*} 6x+2y &= -4\\ 3x+4y &= -17 \end{align*}\)

    23) \(\begin{align*} 2x+3y &= 12\\ 4x+y &= 14 \end{align*}\)

    Answer

    \((3,2)\)

    24) \(\begin{align*} -4x-3y &= -2\\ 3x-5y &= -13 \end{align*}\)

    25) \(\begin{align*} -5x+8y &= 3\\ 10x+6y &= 5 \end{align*}\)

    Answer

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

    26) \(\begin{align*} 3x+4y &= 12\\ -6x-8y &= -24 \end{align*}\)

    27) \(\begin{align*} -60x+45y &= 12\\ 20x-15y &= -4 \end{align*}\)

    Answer

    \(\left (x, \dfrac{4}{15}(5x+1) \right )\)

    28) \(\begin{align*} 11x+10y &= 43\\ 15x+20y &= 65 \end{align*}\)

    29) \(\begin{align*} 2x-y &= 2\\ 3x+2y &= 17 \end{align*}\)

    Answer

    \((3,4)\)

    30) \(\begin{align*} -1.06x-2.25y &= 5.51\\ -5.03x-1.08y &= 5.40 \end{align*}\)

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

    Answer

    \(\left (\dfrac{196}{39}, -\dfrac{5}{13} \right )\)

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

    33) \(\left [ \begin{array}{ccc|c} 1 & 0 & 0 & 31\\ 0 & 1 & 1 & 45\\ 0 & 0 & 1 & 87\\ \end{array} \right ]\)

    Answer

    \((31,-42,87)\)

    34) \(\left [ \begin{array}{ccc|c} 1 & 0 & 1 & 50\\ 1 & 1 & 0 & 20\\ 0 & 1 & 1 & -90\\ \end{array} \right ]\)

    35) \(\left [ \begin{array}{ccc|c} 1 & 2 & 3 & 4\\ 0 & 5 & 6 & 7\\ 0 & 0 & 8 & 9\\ \end{array} \right ]\)

    Answer

    \(\left (\dfrac{21}{40}, \dfrac{1}{20}, \dfrac{9}{8} \right )\)

    36) \(\left [ \begin{array}{ccc|c} -0.1 & 0.3 & -0.1 & 0.2\\ -0.4 & 0.2 & 0.1 & 0.8\\ 0.6 & 0.1 & 0.7 & -0.8\\ \end{array} \right ]\)

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

    Answer

    \(\left (\dfrac{18}{13}, \dfrac{15}{13}, -\dfrac{15}{13} \right )\)

    38) \(\begin{align*} x+y-4z &= -4\\ 5x-3y-2z &= 0\\ 2x+6y+7z &= 30 \end{align*}\)

    39) \(\begin{align*} 2x+3y+2z &= 1\\ -4x-6y-4z &= -2\\ 10x+15y+10z &= 5 \end{align*}\)

    Answer

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

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

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

    Answer

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

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

    43) \(\begin{align*} x+y+z &= 100\\ x+2z &= 125\\ -y+2z &= 25 \end{align*}\)

    Answer

    \((125,-25,0)\)

    44) \(\begin{align*} \dfrac{1}{4}x-\dfrac{2}{3}z &= -\dfrac{1}{2}\\ \dfrac{1}{5}x+\dfrac{1}{3}y &= \dfrac{4}{7}\\ \dfrac{1}{5}y-\dfrac{1}{3}z &= \dfrac{2}{9} \end{align*}\)

    45) \(\begin{align*} -\dfrac{1}{2}x+\dfrac{1}{2}y+\dfrac{1}{7}z &= -\dfrac{53}{14}\\ \dfrac{1}{2}x-\dfrac{1}{2}y+\dfrac{1}{4}z &= 3\\ \dfrac{1}{4}x+\dfrac{1}{5}y+\dfrac{1}{3}z &= \dfrac{23}{15} \end{align*}\)

    Answer

    \((8,1,-2)\)

    46) \(\begin{align*} -\dfrac{1}{2}x-\dfrac{1}{3}y+\dfrac{1}{4}z &= -\dfrac{29}{6}\\ \dfrac{1}{5}x+\dfrac{1}{6}y-\dfrac{1}{7}z &= \dfrac{431}{210}\\ -\dfrac{1}{8}x+\dfrac{1}{9}y+\dfrac{1}{10}z &= -\dfrac{49}{45} \end{align*}\)

    Extensions

    For the exercises 47-51, use Gaussian elimination to solve the system.

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

    Answer

    \((1,2,3)\)

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

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

    Answer

    \(\left (x, \dfrac{31}{28}-\dfrac{3x}{4}, \dfrac{1}{28}(-7x-3) \right )\)

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

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

    Answer

    No solutions exist.

    Real-World Applications

    For the exercises 52-61, set up the augmented matrix that describes the situation, and solve for the desired solution.

    52) Every day, a cupcake store sells \(5,000\) cupcakes in chocolate and vanilla flavors. If the chocolate flavor is \(3\) times as popular as the vanilla flavor, how many of each cupcake sell per day?

    53) At a competing cupcake store, \(\$4,520\) worth of cupcakes are sold daily. The chocolate cupcakes cost \(\$2.25\) and the red velvet cupcakes cost \(\$1.75\). If the total number of cupcakes sold per day is \(2,200\), how many of each flavor are sold each day?

    Answer

    \(860\) red velvet, \(1,340\) chocolate

    54) You invested \(\$10,000\) into two accounts: one that has simple \(3\%\) interest, the other with \(2.5\%\) interest. If your total interest payment after one year was \(\$283.50\), how much was in each account after the year passed?

    55) You invested \(\$2,300\) into account 1, and \(\$2,700\) into account 2. If the total amount of interest after one year is \(\$254\), and account 2 has \(1.5\) times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

    Answer

    \(4\%\) for account 1, \(6\%\) for account 2

    56) Bikes’R’Us manufactures bikes, which sell for \(\$250\). It costs the manufacturer \(\$180\) per bike, plus a startup fee of \(\$3,500\). After how many bikes sold will the manufacturer break even?

    57) A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for \(\$86\) each, with a delivery fee of \(\$9,200\), regardless of how many vacuums are sold. If the store needs to start seeing a profit after \(230\) units are sold, how much should they charge for the vacuums?

    Answer

    \(\$126\)

    58) The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising \(83\%\) of the flavors sold at an ice cream shop. If vanilla sells \(1\%\) more than twice strawberry, and chocolate sells \(11\%\) more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

    59) At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up \(12\%\) of total ice cream sales. This year, the same three ice creams made up \(16.9\%\) of ice cream sales. The rocky road sales doubled, the banana sales increased by \(50\%\), and the pumpkin sales increased by \(20\%\). If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

    Answer

    Banana was \(3\%\), pumpkin was \(7\%\), and rocky road was \(2\%\)

    60) A bag of mixed nuts contains cashews, pistachios, and almonds. There are \(1,000\) total nuts in the bag, and there are \(100\) less almonds than pistachios. The cashews weigh \(3\) g, pistachios weigh \(4\) g, and almonds weigh \(5\) g. If the bag weighs \(3.7\) kg, find out how many of each type of nut is in the bag.

    61) A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were \(900\) nuts in the bag. \(30\%\) of the almonds, \(20\%\) of the cashews, and \(10\%\) of the pistachios were eaten, and now there are \(770\) nuts left in the bag. Originally, there were \(100\) more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

    Answer

    \(100\) almonds, \(200\) cashews, \(600\) pistachios

    9.7: Solving Systems with Inverses

    Verbal

    1) In a previous section, we showed that matrix multiplication is not commutative, that is, \(AB\neq BA\) in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is, \(A^{-1}A=AA^{-1}\)

    Answer

    If \(A^{-1}\) is the inverse of \(A\), then \(AA^{-1}=I\),the identity matrix. Since \(A\) is also the inverse of \(A^{-1}\), \(A^{-1}A=I\). You can also check by proving this for a \(2\times 2\) matrix.

    2) Does every \(2\times 2\) matrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.

    3) Can you explain whether a \(2\times 2\) matrix with an entire row of zeros can have an inverse?

    Answer

    No, because \(ad\) and \(bc\) are both \(0\), so \(ad-bc=0\),which requires us to divide by \(0\) in the formula.

    4) Can a matrix with an entire column of zeros have an inverse? Explain why or why not.

    5) Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a \(2\times 2\) matrix.

    Answer

    Yes. Consider the matrix \(\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\). The inverse is found with the following calculation: 

    \(A^{-1} = \dfrac{1}{0(0)-1(1)} \begin{bmatrix} 0 & -1\\ -1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\)

    Algebraic

    In the exercises 6-12, show that matrix \(A\) is the inverse of matrix \(B\).

    6) \(A = \begin{bmatrix} 1 & 0\\ -1 & 1 \end{bmatrix}, B = \begin{bmatrix} 1 & 0\\ 1 & 1 \end{bmatrix}\)

    7) \(A = \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}, B = \begin{bmatrix} -2 & 1\\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}\)

    Answer

    \(AB = BA = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} = I\)

    8) \(A = \begin{bmatrix} 4 & 5\\ 7 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & \frac{1}{7}\\ \frac{1}{5} & -\frac{4}{35} \end{bmatrix}\)

    9) \(A = \begin{bmatrix} -2 & \frac{1}{2}\\ 3 & -1 \end{bmatrix}, B = \begin{bmatrix} -2 & -1\\ -6 & -4 \end{bmatrix}\)

    Answer

    \(AB = BA = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} = I\)

    10) \(A = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & -1\\ 0 & 1 & 1 \end{bmatrix}, B = \dfrac{1}{2}\begin{bmatrix} 2 & 1 & -1\\ 0 & 1 & 1\\ 0 & -1 & 1 \end{bmatrix}\)

    11) \(A = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & -1\\ 0 & 1 & 1 \end{bmatrix}, B = \dfrac{1}{2}\begin{bmatrix} 2 & 1 & -1\\ 0 & 1 & 1\\ 0 & -1 & 1 \end{bmatrix}\)

    Answer

    \(AB = BA = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I\)

    12) \(A = \begin{bmatrix} 3 & 8 & 2\\ 1 & 1 & 1\\ 5 & 6 & 12 \end{bmatrix}, B = \dfrac{1}{36}\begin{bmatrix} -6 & 84 & -6\\ 7 & -26 & 1\\ -1 & -22 & 5 \end{bmatrix}\)

    For the exercises 13-26, find the multiplicative inverse of each matrix, if it exists.

    13) \(\begin{bmatrix} 3 & -2\\ 1 & 9 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{29}\begin{bmatrix} 9 & 2\\ -1 & 3 \end{bmatrix}\)

    14) \(\begin{bmatrix} -2 & 2\\ 3 & 1 \end{bmatrix}\)

    15) \(\begin{bmatrix} -3 & 7\\ 9 & 2 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{69}\begin{bmatrix} -2 & 7\\ 9 & 3 \end{bmatrix}\)

    16) \(\begin{bmatrix} -4 & -3\\ -5 & 8 \end{bmatrix}\)

    17) \(\begin{bmatrix} 1 & 1\\ 2 & 2 \end{bmatrix}\)

    Answer

    There is no inverse

    18) \(\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\)

    19) \(\begin{bmatrix} 0.5 & 1.5\\ 1 & -0.5 \end{bmatrix}\)

    Answer

    \(\dfrac{4}{7}\begin{bmatrix} 0.5 & 1.5\\ 1 & -0.5 \end{bmatrix}\)

    20) \(\begin{bmatrix} 1 & 0 & 6\\ -2 & 1 & 7\\ 3 & 0 & 2 \end{bmatrix}\)

    21) \(\begin{bmatrix} 0 & 1 & -3\\ 4 & 1 & 0\\ 1 & 0 & 5 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{17}\begin{bmatrix} -5 & 5 & -3\\ 20 & -3 & 12\\ 1 & -1 & 4 \end{bmatrix}\)

    22) \(\begin{bmatrix} 1 & 2 & -1\\ -3 & 4 & 1\\ -2 & -4 & -5 \end{bmatrix}\)

    23) \(\begin{bmatrix} 1 & 9 & -3\\ 2 & 5 & 6\\ 4 & -2 & -7 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{209}\begin{bmatrix} 47 & -57 & 69\\ 10 & 19 & -12\\ -24 & 38 & -13 \end{bmatrix}\)

    24) \(\begin{bmatrix} 1 & -2 & 3\\ -4 & 8 & -12\\ 1 & 4 & 2 \end{bmatrix}\)

    25) \(\begin{bmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5}\\ \frac{1}{6} & \frac{1}{7} & \frac{1}{8} \end{bmatrix}\)

    Answer

    \(\begin{bmatrix} 18 & 60 & -168\\ -56 & -140 & 448\\ 40 & 80 & -280 \end{bmatrix}\)

    26) \(\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{bmatrix}\)

    For the exercises 27-34, solve the system using the inverse of a \(2\times 2\) matrix.

    27) \(\begin{align*} 5x-6y &= -61\\ 4x+3y &= -2 \end{align*}\)

    Answer

    \((-5,6)\)

    28) \(\begin{align*} 8x+4y &= -100\\ 3x-4y &= 1 \end{align*}\)

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

    Answer

    \((2,0)\)

    30) \(\begin{align*} 5x-4y &= -5\\ 4x+y &= 2.3 \end{align*}\)

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

    Answer

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

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

    33) \(\begin{align*} \dfrac{8}{5}x-\dfrac{4}{5}y &= \dfrac{2}{5}\\ -\dfrac{8}{5}x+\dfrac{1}{5}y &= \dfrac{7}{10} \end{align*}\)

    Answer

    \(\left (-\dfrac{2}{3}, -\dfrac{11}{6} \right )\)

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

    For the exercises 35-42, solve a system using the inverse of a \(3\times 3\) matrix.

    35) \(\begin{align*} 3x-2y+5z &= 21\\ 5x+4y &= 37\\ x-2y-5z &= 5 \end{align*}\)

    Answer

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

    36) \(\begin{align*} 4x+4y+4z &= 40\\ 2x-3y+4z &= -12\\ -x+3y+4z &= 9 \end{align*}\)

    37) \(\begin{align*} 6x-5y-z &= 31\\ -x+2y+z &= -6\\ 3x+3y+2z &= 13 \end{align*}\)

    Answer

    \((5,0,-1)\)

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

    39) \(\begin{align*} 4x-2y+3z &= -12\\ 2x+2y-9z &= 33\\ 6y-4z &= 1 \end{align*}\)

    Answer

    \(\dfrac{1}{34} \left(-35, -97, -154 \right)\)

    40) \(\begin{align*} \dfrac{1}{10}x-\dfrac{1}{5}y+4z &= \dfrac{-41}{2}\\ \dfrac{1}{5}x-20y+\dfrac{2}{5}z &= -101\\ \dfrac{3}{10}x+4y-\dfrac{3}{10}z &= 23 \end{align*}\)

    41) \(\begin{align*} \dfrac{1}{2}x-\dfrac{1}{5}y+\dfrac{1}{5}z &= \dfrac{31}{100}\\ -\dfrac{3}{4}x-\dfrac{1}{4}y+\dfrac{1}{2}z &= \dfrac{7}{40}\\ -\dfrac{4}{5}x-\dfrac{1}{2}y+\dfrac{3}{2}z &= \dfrac{1}{4} \end{align*}\)

    Answer

    \(\dfrac{1}{690} \left(65, -1136, -229 \right)\)

    42) \(\begin{align*} 0.1x+0.2y+0.3z &= -1.4\\ 0.1x-0.2y+0.3z &= 0.6\\ 0.4y+0.9z &= -2 \end{align*}\)

    Technology

    For the exercises 43-46, use a calculator to solve the system of equations with matrix inverses.

    43) \(\begin{align*} 2x-y &= -3\\ -x+2y &= 2.3\\ \end{align*}\)

    Answer

    \(\left (-\dfrac{37}{30}, \dfrac{8}{15} \right )\)

    44) \(\begin{align*} -\dfrac{1}{2}x-\dfrac{3}{2}y &= -\dfrac{43}{20}\\ \dfrac{5}{2}x+\dfrac{11}{5}y &= \dfrac{31}{4}\\ \end{align*}\)

    45) \(\begin{align*} 12.3x-2y-2.5z &= 2\\ 36.9x+7y-7.5z &= -7\\ 8y-5z &= -10 \end{align*}\)

    Answer

    \(\left (\dfrac{10}{123}, -1, \dfrac{2}{5} \right )\)

    46) \(\begin{align*} 0.5x-3y+6z &= -0.8\\ 0.7x-2y &= -0.06\\ 0.5x+4y+5z &= 0 \end{align*}\)

    Extensions

    For the exercises 47-51, find the inverse of the given matrix.

    47) \(\begin{bmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 1 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{2}\begin{bmatrix} 2 & 1 & -1 & -1\\ 0 & 1 & 1 & -1\\ 0 & -1 & 1 & 1\\ 0 & 1 & -1 & 1 \end{bmatrix}\)

    48) \(\begin{bmatrix} -1 & 0 & 2 & 5\\ 0 & 0 & 0 & 2\\ 0 & 2 & -1 & 0\\ 1 & -3 & 0 & 1 \end{bmatrix}\)

    49) \(\begin{bmatrix} 1 & -2 & 3 & 0\\ 0 & 1 & 0 & 2\\ 1 & 4 & -2 & 3\\ -5 & 0 & 1 & 1 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{39}\begin{bmatrix} 3 & 2 & 1 & -7\\ 18 & -53 & 32 & 10\\ 24 & -36 & 21 & 9\\ -9 & 46 & -16 & -5 \end{bmatrix}\)

    50) \(\begin{bmatrix} 1 & 2 & 0 & 2 & 3\\ 0 & 2 & 1 & 0 & 0\\ 0 & 0 & 3 & 0 & 1\\ 0 & 2 & 0 & 0 & 1\\ 0 & 0 & 1 & 2 & 0 \end{bmatrix}\)

    51) \(\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix}\)

    Answer

    \(\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ -1 & -1 & -1 & -1 & -1 & 1 \end{bmatrix}\)

    Real-World Applications

    For the exercises 52-61, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.

    52) \(2,400\) tickets were sold for a basketball game. If the prices for floor 1 and floor 2 were different, and the total amount of money brought in is \(\$64,000\), how much was the price of each ticket?

    53) In the previous exercise, if you were told there were \(400\) more tickets sold for floor 2 than floor 1, how much was the price of each ticket?

    Answer

    Infinite solutions.

    54) A food drive collected two different types of canned goods, green beans and kidney beans. The total number of collected cans was \(350\) and the total weight of all donated food was \(348\) lb, \(12\) oz. If the green bean cans weigh \(2\) oz less than the kidney bean cans, how many of each can was donated?

    55) Students were asked to bring their favorite fruit to class. \(95\%\) of the fruits consisted of banana, apple, and oranges. If oranges were twice as popular as bananas, and apples were \(5\%\) less popular than bananas, what are the percentages of each individual fruit?

    Answer

    \(50\%\) oranges, \(25\%\) bananas, \(20\%\) apples

    56) A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at \(\$1\) and the chocolate chip cookies at \(\$0.75\). They raised \(\$700\) and sold \(850\) items. How many brownies and how many cookies were sold?

    57) A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at \(\$13.99\), the beanie at \(\$7.99\), and the cowboy hat at \(\$14.49\). If \(100\) hats were sold this past quarter, \(\$1,119\) was taken in by sales, and the amount of beanies sold was \(10\) more than cowboy hats, how many of each should the clothing store order to replace those already sold?

    Answer

    \(10\) straw hats, \(50\) beanies, \(40\) cowboy hats

    58) Anna, Ashley, and Andrea weigh a combined \(370\) lb. If Andrea weighs \(20\) lb more than Ashley, and Anna weighs \(1.5\) times as much as Ashley, how much does each girl weigh?

    59) Three roommates shared a package of \(12\) ice cream bars, but no one remembers who ate how many. If Tom ate twice as many ice cream bars as Joe, and Albert ate three less than Tom, how many ice cream bars did each roommate eat?

    Answer

    Tom ate \(6\), Joe ate \(3\), and Albert ate \(3\).

    60) A farmer constructed a chicken coop out of chicken wire, wood, and plywood. The chicken wire cost \(\$2\) per square foot, the wood \(\$10\) per square foot, and the plywood \(\$5\) per square foot. The farmer spent a total of \(\$51\), and the total amount of materials used was \(14\) ft2. He used \(3\) ft2  more chicken wire than plywood. How much of each material in did the farmer use?

    61) Jay has lemon, orange, and pomegranate trees in his backyard. An orange weighs \(8\) oz, a lemon \(5\) oz, and a pomegranate \(11\) oz. Jay picked \(142\) pieces of fruit weighing a total of \(70\) lb, \(10\) oz. He picked \(15.5\) times more oranges than pomegranates. How many of each fruit did Jay pick?

    Answer

    \(124\) oranges, \(10\) lemons, \(8\) pomegranates 

    9.8: Solving Systems with Cramer's Rule

    Verbal

    1) Explain why we can always evaluate the determinant of a square matrix.

    Answer

    A determinant is the sum and products of the entries in the matrix, so you can always evaluate that product—even if it does end up being \(0\).

    2) Examining Cramer’s Rule, explain why there is no unique solution to the system when the determinant of your matrix is \(0\). For simplicity, use a \(2\times 2\) matrix.

    3) Explain what it means in terms of an inverse for a matrix to have a \(0\) determinant.

    Answer

    The inverse does not exist.

    4) The determinant of \(2\times 2\) matrix \(A\) is \(3\). If you switch the rows and multiply the first row by \(6\) and the second row by \(2\), explain how to find the determinant and provide the answer.

    Algebraic

    For the exercises 5-24, find the determinant.

    5) \(\begin{vmatrix} 1 & 2\\ 3 & 4 \end{vmatrix}\)

    Answer

    \(-2\)

    6) \(\begin{vmatrix} -1 & 2\\ 3 & -4 \end{vmatrix}\)

    7) \(\begin{vmatrix} 2 & -5\\ -1 & 6 \end{vmatrix}\)

    Answer

    \(7\)

    8) \(\begin{vmatrix} -8 & 4\\ -1 & 5 \end{vmatrix}\)

    9) \(\begin{vmatrix} 1 & 0\\ 3 & -4 \end{vmatrix}\)

    Answer

    \(-4\)

    10) \(\begin{vmatrix} 10 & 20\\ 0 & -10 \end{vmatrix}\)

    11) \(\begin{vmatrix} 10 & 0.2\\ 5 & 0.1 \end{vmatrix}\)

    Answer

    \(0\)

    12) \(\begin{vmatrix} 6 & -3\\ 8 & 4 \end{vmatrix}\)

    13) \(\begin{vmatrix} -2 & -3\\ 3.1 & 4,000 \end{vmatrix}\)

    Answer

    \(-7,990.7\)

    14) \(\begin{vmatrix} -1.1 & 0.6\\ 7.2 & -0.5 \end{vmatrix}\)

    15) \(\begin{vmatrix} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -3 \end{vmatrix}\)

    Answer

    \(3\)

    16) \(\begin{vmatrix} -1 & 4 & 0\\ 0 & 2 & 3\\ 0 & 0 & -3 \end{vmatrix}\)

    17) \(\begin{vmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{vmatrix}\)

    Answer

    \(-1\)

    18) \(\begin{vmatrix} 2 & -3 & 1\\ 3 & -4 & 1\\ -5 & 6 & 1 \end{vmatrix}\)

    19) \(\begin{vmatrix} -2 & 1 & 4\\ -4 & 2 & -8\\ 2 & -8 & -3 \end{vmatrix}\)

    Answer

    \(224\)

    20) \(\begin{vmatrix} 6 & -1 & 2\\ -4 & -3 & 5\\ 1 & 9 & -1 \end{vmatrix}\)

    21) \(\begin{vmatrix} 5 & 1 & -1\\ 2 & 3 & 1\\ 3 & -6 & -3 \end{vmatrix}\)

    Answer

    \(15\)

    22) \(\begin{vmatrix} 1.1 & 2 & -1\\ -4 & 0 & 0\\ 4.1 & -0.4 & 2.5 \end{vmatrix}\)

    23) \(\begin{vmatrix} 2 & -1.6 & 3.1\\ 1.1 & 3 & -8\\ -9.3 & 0 & 2 \end{vmatrix}\)

    Answer

    \(-17.03\)

    24) \(\begin{vmatrix} -\frac{1}{2} & \frac{1}{3} & \frac{1}{4}\\ \frac{1}{5} & -\frac{1}{6} & \frac{1}{7}\\ 0 & 0 & \frac{1}{8} \end{vmatrix}\)

    For the exercises 25-34, solve the system of linear equations using Cramer’s Rule.

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

    Answer

    \((1,1)\)

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

    27) \(\begin{align*} 6x-3y &= 2\\ -8x+9y &= -1 \end{align*}\)

    Answer

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

    28) \(\begin{align*} 2x+6y &= 12\\ 5x-2y &= 13 \end{align*}\)

    29) \(\begin{align*} 4x+3y &= 23\\ 2x-y &= -1 \end{align*}\)

    Answer

    \((2,5)\)

    30) \(\begin{align*} 10x-6y &= 2\\ -5x+8y &= -1 \end{align*}\)

    31) \(\begin{align*} 4x-3y &= -3\\ 2x+6y &= -4 \end{align*}\)

    Answer

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

    32) \(\begin{align*} 4x-5y &= 7\\ -3x+9y &= 0 \end{align*}\)

    33) \(\begin{align*} 4x+10y &= 180\\ -3x-5y &= -105 \end{align*}\)

    Answer

    \((15,12)\)

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

    For the exercises 35-44, solve the system of linear equations using Cramer’s Rule.

    35) \(\begin{align*} x+2y-4z &= -1\\ 7x+3y+5z &= 26\\ -2x-6y+7z &= -6 \end{align*}\)

    Answer

    \((1,3,2)\)

    36) \(\begin{align*} -5x+2y-4z &= -47\\ 4x-3y-z &= -94\\ 3x-3y+2z &= 94 \end{align*}\)

    37) \(\begin{align*} 4x+5y-z &= -7\\ -2x-9y+2z &= 8\\ 5y+7z &= 21 \end{align*}\)

    Answer

    \((-1,0,3)\)

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

    39) \(\begin{align*} 4x-2y+3z &= 6\\ -6x+y &= -2\\ 2x+7y+8z &= 24 \end{align*}\)

    Answer

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

    40) \(\begin{align*} 5x+2y-z &= 1\\ -7x-8y+3z &= 1.5\\ 6x-12y+z &= 7 \end{align*}\)

    41) \(\begin{align*} 13x-17y+16z &= 73\\ -11x+15y+17z &= 61\\ 46x+10y-30z &= -18 \end{align*}\)

    Answer

    \((2,1,4)\)

    42) \(\begin{align*} -4x-3y-8z &= -7\\ 2x-9y+5z &= 0.5\\ 5x-6y-5z &= -2 \end{align*}\)

    43) \(\begin{align*} 4x-6y+8z &= 10\\ -2x+3y-4z &= -5\\ x+y+z &= 1 \end{align*}\)

    Answer

    Infinite solutions

    44) \(\begin{align*} 4x-6y+8z &= 10\\ -2x+3y-4z &= -5\\ 12x+18y-24z &= -30 \end{align*}\)

    Technology

    For the exercises 45-48, use the determinant function on a graphing utility.

    45) \(\begin{vmatrix} 1 & 0 & 8 & 9\\ 0 & 2 & 1 & 0\\ 1 & 0 & 3 & 0\\ 0 & 2 & 4 & 3 \end{vmatrix}\)

    Answer

    \(24\)

    46) \(\begin{vmatrix} 1 & 0 & 2 & 1\\ 0 & -9 & 1 & 3\\ 3 & 0 & -2 & -1\\ 0 & 1 & 1 & -2 \end{vmatrix}\)

    47) \(\begin{vmatrix} \frac{1}{2} & 1 & 7 & 4\\ 0 & \frac{1}{2} & 100 & 5\\ 0 & 0 & 2 & 2,000\\ 0 & 0 & 0 & 2 \end{vmatrix}\)

    Answer

    \(1\)

    48) \(\begin{vmatrix} 1 & 0 & 0 & 0\\ 2 & 3 & 0 & 0\\ 4 & 5 & 6 & 0\\ 7 & 8 & 9 & 0 \end{vmatrix}\)

    Real-World Applications

    For the exercises 49-52, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.

    49) Two numbers add up to \(56\). One number is \(20\) less than the other.

    Answer

    Yes; \(18\), \(38\)

    50) Two numbers add up to \(104\). If you add two times the first number plus two times the second number, your total is \(208\)

    51) Three numbers add up to \(106\). The first number is \(3\) less than the second number. The third number is \(4\) more than the first number.

    Answer

    Yes; \(33\), \(36\), \(37\)

    52) Three numbers add to \(216\). The sum of the first two numbers is \(112\). The third number is 8 less than the first two numbers combined.

    For the exercises 53-65, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.

    53) You invest \(\$10,000\) into two accounts, which receive \(8\%\) interest and \(5\%\) interest. At the end of a year, you had \(\$10,710\) in your combined accounts. How much was invested in each account?

    Answer

    \(\$7,000\) in first account, \(\$3,000\) in second account.

    54) You invest \(\$80,000\) into two accounts, \(\$22,000\) in one account, and \(\$58,000\) in the other account. At the end of one year, assuming simple interest, you have earned \(\$2,470\) in interest. The second account receives half a percent less than twice the interest on the first account. What are the interest rates for your accounts?

    55) A movie theater needs to know how many adult tickets and children tickets were sold out of the \(1,200\) total tickets. If children’s tickets are \(\$5.95\), adult tickets are \(\$11.15\), and the total amount of revenue was \(\$12,756\), how many children’s tickets and adult tickets were sold?

    Answer

    \(120\) children, \(1,080\) adult

    56) A concert venue sells single tickets for \(\$40\) each and couple’s tickets for \(\$65\). If the total revenue was \(\$18,090\) and the \(321\) tickets were sold, how many single tickets and how many couple’s tickets were sold?

    57) You decide to paint your kitchen green. You create the color of paint by mixing yellow and blue paints. You cannot remember how many gallons of each color went into your mix, but you know there were \(10\) gal total. Additionally, you kept your receipt, and know the total amount spent was \(\$29.50\). If each gallon of yellow costs \(\$2.59\), and each gallon of blue costs \(\$3.19\), how many gallons of each color go into your green mix?

    Answer

    \(4\) gal yellow, \(6\) gal blue

    58) You sold two types of scarves at a farmers’ market and would like to know which one was more popular. The total number of scarves sold was \(56\), the yellow scarf cost \(\$10\), and the purple scarf cost \(\$11\). If you had total revenue of \(\$583\), how many yellow scarves and how many purple scarves were sold?

    59) Your garden produced two types of tomatoes, one green and one red. The red weigh \(10\) oz, and the green weigh \(4\) oz. You have \(30\) tomatoes, and a total weight of \(13\) lb, \(14\) oz. How many of each type of tomato do you have?

    Answer

    \(13\) green tomatoes, \(17\) red tomatoes

    60) At a market, the three most popular vegetables make up \(53\%\) of vegetable sales. Corn has \(4\%\) higher sales than broccoli, which has \(5\%\) more sales than onions. What percentage does each vegetable have in the market share?

    61) At the same market, the three most popular fruits make up \(37\%\) of the total fruit sold. Strawberries sell twice as much as oranges, and kiwis sell one more percentage point than oranges. For each fruit, find the percentage of total fruit sold.

    Answer

    Strawberries \(18\%\), oranges \(9\%\), kiwi \(10\%\)

    62) Three bands performed at a concert venue. The first band charged \(\$15\) per ticket, the second band charged \(\$45\) per ticket, and the final band charged \(\$22\) per ticket. There were \(510\) tickets sold, for a total of \(\$12,700\). If the first band had \(40\) more audience members than the second band, how many tickets were sold for each band?

    63) A movie theatre sold tickets to three movies. The tickets to the first movie were \(\$5\), the tickets to the second movie were \(\$11\), and the third movie was \(\$12\). \(100\) tickets were sold to the first movie. The total number of tickets sold was \(642\), for a total revenue of \(\$6,774\). How many tickets for each movie were sold?

    Answer

    \(100\) for movie 1, \(230\) for movie 2, \(312\) for movie 3

    64) Men aged \(20–29\), \(30–39\), and \(40–49\) made up \(78\%\) of the population at a prison last year. This year, the same age groups made up \(82.08\%\) of the population. The \(20–29\) age group increased by \(20%\), the \(30–39\) age group increased by \(2\%\), and the \(40–49\) age group decreased to \(\dfrac{3}{4}\) of their previous population. Originally, the \(30–39\) age group had \(2\%\) more prisoners than the \(20–29\) age group. Determine the prison population percentage for each age group last year.

    65) At a women’s prison down the road, the total number of inmates aged \(20–49\) totaled \(5,525\). This year, the \(20–29\) age group increased by \(10\%\), the \(30–39\) age group decreased by \(20\%\), and the \(40–49\) age group doubled. There are now \(6,040\) prisoners. Originally, there were \(500\) more in the \(30–39\) age group than the \(20–29\) age group. Determine the prison population for each age group last year.

    Answer

    \(20–29: 2,100\), \(30–39: 2,600\), \(40–49: 825\)

    For the exercises 66-68, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in the Table below.

      Fat (g) Protein (g) Carbohydrates (g)
    Almonds (10) 6 2 3
    Cranberries (10) 0.02 0 8
    Cashews (10) 7 3.5 5.5

    66) For the special “low-carb”trail mix, there are \(1,000\) pieces of mix. The total number of carbohydrates is \(425\) g, and the total amount of fat is \(570.2\) g. If there are \(200\) more pieces of cashews than cranberries, how many of each item is in the trail mix?

    67) For the “hiking” mix, there are \(1,000\) pieces in the mix, containing \(390.8\) g of fat, and \(165\) g of protein. If there is the same amount of almonds as cashews, how many of each item is in the trail mix?

    Answer

    \(300\) almonds, \(400\) cranberries, \(300\) cashews

    68) For the “energy-booster” mix, there are \(1,000\) pieces in the mix, containing \(145\) g of protein and \(625\) g of carbohydrates. If the number of almonds and cashews summed together is equivalent to the amount of cranberries, how many of each item is in the trail mix?

    Contributors