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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

    2) \(\begin{align*} y &= x^2 - 4\\ y &= 5x+10 \end{align*}\)

    3) \(\begin{align*} x^2 + y^2 &= 16\\ y &= x-8 \end{align*}\)

    Answer

    No solution

    4) \(\begin{align*} x^2 + y^2 &= 25\\ y &= x^2 + 5 \end{align*}\)

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

    Answer

    No solution

    For the exercises 6-7, graph the inequality.

    6) \(y>x^2 - 1\)

    7) \(\dfrac{1}{4}x^2 + y^2 < 4\)

    Answer

    CNX_Precalc_Figure_09_08_202.jpg

    For the exercises 8-10, graph the system of inequalities.

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

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

    Answer

    CNX_Precalc_Figure_09_08_204.jpg

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

    9.4: Partial Fractions

    For the exercises 1-8, decompose into partial fractions.

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

    Answer

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

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

    3) \(\dfrac{7x+20}{x^2 +10x+25}\)

    Answer

    \(\dfrac{7}{x+5}, \dfrac{-15}{(x+5)^2}\)

    4) \(\dfrac{x-18}{x^2 -12x+36}\)

    5) \(\dfrac{-x^2 +36x + 70}{x^3 -125}\)

    Answer

    \(\dfrac{3}{x-5}, \dfrac{-4x+1}{x^2 +5x+25}\)

    6) \(\dfrac{-5x^2 +6x-2}{x^3 +27}\)

    7) \(\dfrac{x^3 -4x^2 +3x+11}{(x^2 -2)^2}\)

    Answer

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

    8) \(\dfrac{4x^4 -2x^3 +22x^2 -6x+48}{x(x^2 +4)^2}\)

    9.5: Matrices and Matrix Operations

    For the exercises 1-12, perform the requested operations on the given matrices.

    \[A=\begin{bmatrix} 4 & -2\\ 1 & 3 \end{bmatrix}, \begin{bmatrix} 6 & 7 & -3\\ 11 & -2 & 4 \end{bmatrix}, C=\begin{bmatrix} 6 & 7\\ 11 & -2\\ 14 & 0 \end{bmatrix} D=\begin{bmatrix} 1 & -4 & 9\\ 10 & 5 & -7\\ 2 & 8 & 5 \end{bmatrix} E=\begin{bmatrix} 7 & -14 & 3\\ 2 & -1 & 3\\ 0 & 1 & 9 \end{bmatrix} \nonumber\]

    1) \(-4A\)

    Answer

    \(\begin{bmatrix} -16 & 8\\ -4 & -12 \end{bmatrix}\)

    2) \(10D-6E\)

    3) \(B+C\)

    Answer

    undefined; dimensions do not match

    4) \(AB\)

    5) \(BA\)

    Answer

    undefined; inner dimensions do not match

    6) \(BC\)

    7) \(CB\)

    Answer

    \(\begin{bmatrix} 113 & 28 & 10\\ 44 & 81 & -41\\ 84 & 98 & -42 \end{bmatrix}\)

    8) \(DE\)

    9) \(ED\)

    Answer

    \(\begin{bmatrix} -127 & -74 & 176\\ -2 & 11 & 40\\ 28 & 77 & 38 \end{bmatrix}\)

    10) \(EC\)

    11) \(CE\)

    Answer

    undefined; inner dimensions do not match

    12) \(A^3\)

    9.6: Solving Systems with Gaussian Elimination

    For the exercises 1-2, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

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

    Answer

    \(\begin{align*} x-3z &= 7\\ y+2z &= -5 \end{align*}\; \; \text{with infinite solutions}\)

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

    For the exercises 3-5, write the augmented matrix from the system of linear equations.

    3) \(\begin{align*} -2x+2y+z &= 7\\ 2x-8y+5z &= 0\\ 19x-10y+22z &= 3 \end{align*}\)

    Answer

    \(\left [ \begin{array}{ccc|c} -2 & 2 & 1 & 7 \\ 2 & -8 & 5 & 0\\ 19 & -10 & 22 & 3\\ \end{array} \right ]\)

    4) \(\begin{align*} 4x+2y-3z &= 14\\ -12x+3y+z &= 100\\ 9x-6y+2z &= 31 \end{align*}\)

    5) \(\begin{align*} x+3z &= 12\\ -x+4y &= 0\\ y+2z &= -7 \end{align*}\)

    Answer

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

    For the exercises 6-10, solve the system of linear equations using Gaussian elimination.

    6) \(\begin{align*} 3x-4y &= -7\\ -6x+8y &= 14 \end{align*}\)

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

    Answer

    No solutions exist.

    8) \(\begin{align*} -1.1x-2.3y &= 6.2\\ -5.2x-4.1y &= 4.3 \end{align*}\)

    9) \(\begin{align*} 2x+3y+2z &= 1\\ -4x-6y-4z &= -2\\ 10x+15y+10z &= 0 \end{align*}\)

    Answer

    No solutions exist.

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

    9.7: Solving Systems with Inverses

    For the exercises 1-4, find the inverse of the matrix.

    1) \(\begin{bmatrix} -0.2 & 1.4\\ 1.2 & -0.4 \end{bmatrix}\)

    Answer

    \(\dfrac{1}{8}\begin{bmatrix} 2 & 7\\ 6 & 1 \end{bmatrix}\)

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

    3) \(\begin{bmatrix} 12 & 9 & -6\\ -1 & 3 & 2\\ -4 & -3 & 2 \end{bmatrix}\)

    Answer

    No inverse exists.

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

    For the exercises 5-8, find the solutions by computing the inverse of the matrix.

    5) \(\begin{align*} 0.3x-0.1y &= -10\\ -0.1x+0.3y &= 14 \end{align*}\)

    Answer

    \((-20,40)\)

    6) \(\begin{align*} 0.4x-0.2y &= -0.6\\ -0.1x+0.05y &= 0.3 \end{align*}\)

    7) \(\begin{align*} 4x+3y-3z &= -4.3\\ 5x-4y-z &= -6.1\\ x+z &= -0.7 \end{align*}\)

    Answer

    \((-1, 0.2, 0.3)\)

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

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

    9) Students were asked to bring their favorite fruit to class. \(90\%\) of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were \(5\%\) more popular than bananas, what are the percentages of each individual fruit?

    Answer

    \(17\%\) oranges, \(34\%\) bananas, \(39\%\) apples

    10) A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at \(\$2\) and the chocolate chip cookies at \(\$1\). They raised \(\$250\) and sold \(175\) items. How many brownies and how many cookies were sold?

    9.8: Solving Systems with Cramer's Rule

    For the exercises 1-4, find the determinant.

    1) \(\begin{vmatrix} 100 & 0\\ 0 & 0 \end{vmatrix}\)

    Answer

    \(0\)

    2) \(\begin{vmatrix} 0.2 & -0.6\\ 0.7 & -1.1 \end{vmatrix}\)

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

    Answer

    \(6\)

    4) \(\begin{vmatrix} \sqrt{2} & 0 & 0\\ 0 & \sqrt{2} & 0\\ 0 & 0 & \sqrt{2} \end{vmatrix}\)

    For the exercises 5-10, use Cramer’s Rule to solve the linear systems of equations.

    5) \(\begin{align*} 4x-2y &= 23\\ -5x-10y &= -35 \end{align*}\)

    Answer

    \(\left(6, \dfrac{1}{2} \right)\)

    6) \(\begin{align*} 0.2x-0.1y &= 0\\ -0.3x+0.3y &= 2.5 \end{align*}\)

    7) \(\begin{align*} -0.5x+0.1y &= 0.3\\ -0.25x+0.05y &= 0.15 \end{align*}\)

    Answer

    \(x, 5x+3\)

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

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

    Answer

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

    10) \(\begin{align*} \dfrac{3}{10}x-\dfrac{1}{5}y-\dfrac{3}{10}z &= -\dfrac{1}{50}\\ \dfrac{1}{10}x-\dfrac{1}{10}y-\dfrac{1}{2}z &= -\dfrac{9}{50}\\ \dfrac{2}{5}x-\dfrac{1}{2}y-\dfrac{3}{5}z &= -\dfrac{1}{5} \end{align*}\)

    Practice Test

    1) Is the following ordered pair a solution to the system of equations? \[\begin{align*} -5x-y &= 12 \text{ with } (-3,3)\\ x+4y &= 9 \end{align*} \nonumber \]

    Answer

    Yes

    For the exercises 2-9, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

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

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

    Answer

    No solutions exist.

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

    5) \(\begin{align*} 4x-6y-2z &= \dfrac{1}{10}\\ x-7y+5z &= -\dfrac{1}{4}\\ 3x+6y-9z &= \dfrac{6}{5} \end{align*}\)

    Answer

    \(\dfrac{1}{20} (10, 5, 4)\)

    6) \(\begin{align*} x+z &= 20\\ x+y+z &= 20\\ x+2y+z &= 10 \end{align*}\)

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

    Answer

    \(\left ( x, \dfrac{16x}{5} - \dfrac{13x}{5} \right )\)

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

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

    Answer

    \((-2\sqrt{2}, -\sqrt{17}), (-2\sqrt{2}, \sqrt{17}), (2\sqrt{2}, -\sqrt{17}), (2\sqrt{2}, \sqrt{17})\)

    For the exercises 10-11, graph the following inequalities.

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

    11) \(\begin{align*} x^2 + y^2 &> 4 \\ y &< x^2 + 1 \end{align*}\)

    Answer

    9PT11.png

    For the exercises 12-14, write the partial fraction decomposition.

    12) \(\dfrac{-8x-30}{x^2 + 10x+25}\)

    13) \(\dfrac{13x+2}{(3x+1)^2}\)

    Answer

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

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

    For the exercises 15-21, perform the given matrix operations.

    15) \(5\begin{bmatrix} 4 & 9\\ -2 & 3 \end{bmatrix}+\dfrac{1}{2} \begin{bmatrix} -6 & 12\\ 4 & -8 \end{bmatrix}\)

    Answer

    \(\begin{bmatrix} 17 & 51\\ -8 & 11 \end{bmatrix}\)

    16) \(\begin{bmatrix} 1 & 4 & -7\\ -2 & 9 & 5\\ 12 & 0 & -4 \end{bmatrix} \begin{bmatrix} 3 & -4\\ 1 & 3\\ 5 & 10 \end{bmatrix}\)

    17) \(\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{4} & \frac{1}{5} \end{bmatrix} ^{-1}\)

    Answer

    \(\begin{bmatrix} 12 & -20\\ -15 & 30 \end{bmatrix}\)

    18) \(\textbf{det}\begin{vmatrix} 0 & 0\\ 400 & 4,000 \end{vmatrix}\)

    19) \(\textbf{det}\begin{vmatrix} \frac{1}{2} & -\frac{1}{2} & 0\\ -\frac{1}{2} & 0 & \frac{1}{2}\\ 0 & \frac{1}{2} & 0 \end{vmatrix}\)

    Answer

    \(-\dfrac{1}{8}\)

    20) If \(\textbf{det}(A)=-6\), what would be the determinant if you switched rows 1 and 3, multiplied the second row by \(12\), and took the inverse?

    21) Rewrite the system of linear equations as an augmented matrix. \[\begin{align*} 14x-2y-13z &= 140\\ -2x+3y-6z &= -1\\ x-5y+12z &= 11 \end{align*} \nonumber\]

    Answer

    \(\left [ \begin{array}{ccc|c} 14 & -2 & 13 & 140 \\ -2 & 3 & -6 & -1\\ 1 & -5 & 12 & 11\\ \end{array} \right ]\)

    22) Rewrite the augmented matrix as a system of linear equations. \[\left [ \begin{array}{ccc|c} 1 & 0 & 3 & 12 \\ -2 & 4 & 9 & -5\\ -6 & 1 & 2 & 8\\ \end{array} \right ] \nonumber\]

    For the exercises 23-24, use Gaussian elimination to solve the systems of equations.

    23) \(\begin{align*} x-6y &= 4\\ 2x-12y &= 0 \end{align*}\)

    Answer

    No solutions exist.

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

    For the exercises 25-26, use the inverse of a matrix to solve the systems of equations.

    25) \(\begin{align*} 4x-5y &= -50\\ -x+2y &= 80 \end{align*}\)

    Answer

    \((100, 90)\)

    26) \(\begin{align*} \dfrac{1}{100}x-\dfrac{3}{100}y+\dfrac{1}{20}z &= -49\\ \dfrac{3}{100}x-\dfrac{7}{100}y-\dfrac{1}{100}z &= 13\\ \dfrac{9}{100}x-\dfrac{9}{100}y-\dfrac{9}{100}z &= 99 \end{align*}\)

    For the exercises 27-28, use Cramer’s Rule to solve the systems of equations.

    27) \(\begin{align*} 200x-300y &= 2\\ 400x+715y &= 4 \end{align*}\)

    Answer

    \(\left (\dfrac{1}{100}, 0 \right )\)

    28) \(\begin{align*} 0.1x+0.1y-0.1z &= -1.2\\ 0.1x-0.2y+0.4z &= -1.2\\ 0.5x-0.3y+0.8z &= -5.9 \end{align*}\)

    For the exercises 29-30, solve using a system of linear equations.

    29) A factory producing cell phones has the following cost and revenue functions: \(C(x)=x^2+75x+2,688\) and \(R(x)=x^2+160x\). What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

    Answer

    \(32\) or more cell phones per day

    30) A small fair charges \(\$1.50\) for students, \(\$1\) for children, and \(\$2\) for adults. In one day, three times as many children as adults attended. A total of \(800\) tickets were sold for a total revenue of \(\$1,050\). How many of each type of ticket was sold?

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