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Mathematics LibreTexts

9.R: Systems of Equations and Inequalities (Review)

  • Page ID
    18812
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    9.1: Systems of Linear Equations: Two Variables

    For the exercises 1-2, determine whether the ordered pair is a solution to the system of equations.

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

    Answer

    No

    2) \(\begin{align*} 6x-2y &= 24\\ -3x+3y &= 18 \end{align*}\; \; \text{ and }\; (9,15)\)

    For the exercises 3-5, use substitution to solve the system of equations.

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

    Answer

    \((-2,3)\)

    4) \(\begin{align*} \dfrac{4}{7}x+\dfrac{1}{5}y &= \dfrac{43}{70}\\ \dfrac{5}{6}x-\dfrac{1}{3}y &= -\dfrac{2}{3} \end{align*}\)

    5) \(\begin{align*} 5x+6y &= 14\\ 4x+8y &= 8 \end{align*}\)

    Answer

    \((4,-1)\)

    For the exercises 6-8, use addition to solve the system of equations.

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

    7) \(\begin{align*} 3x+4y &= 2\\ 9x+12y &= 3 \end{align*}\)

    Answer

    No solutions exist.

    8) \(\begin{align*} 8x+4y &= 2\\ 6x-5y &= 0.7 \end{align*}\)

    For the exercises 9-10, write a system of equations to solve each problem. Solve the system of equations.

    9) A factory has a cost of production \(C(x)=150x+15,000\) and a revenue function \(R(x)=200x\). What is the break-even point?

    Answer

    \((300,60,000)\)

    10) A performer charges \(C(x)=50x+10,000\), where \(x\) is the total number of attendees at a show. The venue charges \(\$75\) 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

    \((400,30,000)\)

    9.2: Systems of Linear Equations: Three Variables

    For the exercises 1-8, solve the system of three equations using substitution or addition.

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

    Answer

    \((10,-10,10)\)

    2) \(\begin{align*} 5x+3y-z &= 5\\ 3x-2y+4z &= 13\\ 4x+3y+5z &= 22 \end{align*}\)

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

    Answer

    No solutions exist.

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

    5) \(\begin{align*} 3x+2y-z &= -10\\ x-y+2z &= 7\\ -x+3y+z &= -2 \end{align*}\)

    Answer

    \((-1,-2,3)\)

    6) \(\begin{align*} 3x+4z &= -11\\ x-2y &= 5\\ 4y-z &= -10 \end{align*}\)

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

    Answer

    \(\left (x, \dfrac{8x}{5}, \dfrac{14x}{5} \right )\)

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

    For the exercises 9-10, write a system of equations to solve each problem. Solve the system of equations.

    9) Three odd numbers sum up to \(61\). The smaller is one-third the larger and the middle number is \(16\) less than the larger. What are the three numbers?

    Answer

    \(11, 17, 33\)

    10) A local theatre sells out for their show. They sell all \(500\) tickets for a total purse of \(\$8,070.00\). The tickets were priced at \(\$15\) for students, \(\$12\) for children, and \(\$18\) for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

    9.3: Systems of Nonlinear Equations and Inequalities: Two Variables

    For the exercises 1-5, solve the system of nonlinear equations.

    1) \(\begin{align*} y &= x^2 - 7\\ y &= 5x-13 \end{align*}\)

    Answer

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