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$\begin{array}{l}\text{}0.5x-0.5y=10\hfill \\ \text{}-0.2y+0.2x=4\hfill \\ \text{}0.1x+0.1z=2\hfill \end{array}$
$\left(10,-10,10\right)$
$\begin{array}{r}\hfill 5x+3y-z=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 3x-2y+4z=13\\ \hfill 4x+3y+5z=22\end{array}$
$\begin{array}{r}x+y+z=1\\ 2x+2y+2z=1\\ 3x+3y=2\end{array}$
No solutions exist.
$\begin{array}{l}\text{}2x-3y+z=\mathrm{-1}\hfill \\ \text{}x+y+z=\mathrm{-4}\hfill \\ \text{}4x+2y-3z=33\hfill \end{array}$
$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3x+2y-z=\mathrm{-10}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-y+2z=7\hfill \\ -x+3y+z=\mathrm{-2}\hfill \end{array}$
$\left(-1,-2,3\right)$
$\begin{array}{r}\hfill 3x+4z=\mathrm{-11}\\ \hfill x-2y=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 4y-z=\mathrm{-10}\end{array}$
$\begin{array}{r}2x-3y+z=0\\ 2x+4y-3z=0\\ 6x-2y-z=0\end{array}$
$\left(x,\frac{8x}{5},\frac{14x}{5}\right)$
$\begin{array}{r}6x-4y-2z=2\\ 3x+2y-5z=4\\ 6y-7z=5\end{array}$
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.
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?
11, 17, 33
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?
For the following exercises, solve the system of nonlinear equations.
$\begin{array}{l}\begin{array}{l}\\ y={x}^{2}-7\end{array}\hfill \\ y=5x-13\hfill \end{array}$
$\left(2,-3\right),\left(3,2\right)$
$\begin{array}{l}\begin{array}{l}\\ y={x}^{2}-4\end{array}\hfill \\ y=5x+10\hfill \end{array}$
$\begin{array}{l}{x}^{2}+{y}^{2}=16\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=x-8\hfill \end{array}$
No solution
$\begin{array}{l}{x}^{2}+{y}^{2}=25\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y={x}^{2}+5\hfill \end{array}$
$\begin{array}{r}{x}^{2}+{y}^{2}=4\\ y-{x}^{2}=3\end{array}$
No solution
For the following exercises, graph the inequality.
$y>{x}^{2}-1$
For the following exercises, graph the system of inequalities.
$\begin{array}{l}{x}^{2}+{y}^{2}+2x<3\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y>-{x}^{2}-3\hfill \end{array}$
$\begin{array}{l}{x}^{2}-2x+{y}^{2}-4x<4\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y<-x+4\hfill \end{array}$
$\begin{array}{l}{x}^{2}+{y}^{2}<1\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}^{2}<x\hfill \end{array}$
For the following exercises, decompose into partial fractions.
$\frac{10x+2}{4{x}^{2}+4x+1}$
$\frac{7x+20}{{x}^{2}+10x+25}$
$\frac{7}{x+5},\frac{-15}{{(x+5)}^{2}}$
$\frac{x-18}{{x}^{2}-12x+36}$
$\frac{-{x}^{2}+36x+70}{{x}^{3}-125}$
$\frac{3}{x-5},\frac{-4x+1}{{x}^{2}+5x+25}$
$\frac{-5{x}^{2}+6x-2}{{x}^{3}+27}$
$\frac{{x}^{3}-4{x}^{2}+3x+11}{{({x}^{2}-2)}^{2}}$
$\frac{x-4}{({x}^{2}-2)},\frac{5x+3}{{({x}^{2}-2)}^{2}}$
$\frac{4{x}^{4}-2{x}^{3}+22{x}^{2}-6x+48}{x{({x}^{2}+4)}^{2}}$
For the following exercises, perform the requested operations on the given matrices.
$-4A$
$\left[\begin{array}{cc}-16& 8\\ -4& -12\end{array}\right]$
$CB$
$\left[\begin{array}{ccc}113& 28& 10\\ 44& 81& -41\\ 84& 98& -42\end{array}\right]$
$ED$
$\left[\begin{array}{ccc}-127& -74& 176\\ -2& 11& 40\\ 28& 77& 38\end{array}\right]$
${A}^{3}$
For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.
$\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill \mathrm{-3}\\ \hfill 0& \hfill 1& \hfill 2\\ \hfill 0& \hfill 0& \hfill 0\end{array}\text{}|\text{}\begin{array}{r}\hfill 7\\ \hfill \mathrm{-5}\\ \hfill 0\end{array}\right]$
$\begin{array}{l}x-3z=7\\ y+2z=-5\text{\hspace{0.17em}}\end{array}$ with infinite solutions
$\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 5\\ \hfill 0& \hfill 1& \hfill \mathrm{-2}\\ \hfill 0& \hfill 0& \hfill 0\end{array}\text{}|\text{}\begin{array}{r}\hfill \mathrm{-9}\\ \hfill 4\\ \hfill 3\end{array}\right]$
For the following exercises, write the augmented matrix from the system of linear equations.
$\begin{array}{l}\\ \begin{array}{r}\hfill -2x+2y+z=7\\ \hfill 2x-8y+5z=0\\ \hfill 19x-10y+22z=3\end{array}\end{array}$
$\left[\begin{array}{rrr}\hfill -2& \hfill 2& \hfill 1\\ \hfill 2& \hfill -8& \hfill 5\\ \hfill 19& \hfill -10& \hfill 22\end{array}\text{}|\text{}\begin{array}{r}\hfill 7\\ \hfill 0\\ \hfill 3\end{array}\right]$
$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4x+2y-3z=14\hfill \\ -12x+3y+z=100\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9x-6y+2z=31\hfill \end{array}$
$\begin{array}{r}\hfill x+3z=12\text{\hspace{0.17em}}\\ \hfill -x+4y=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill y+2z=-7\end{array}$
$\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 3\\ \hfill \mathrm{-1}& \hfill 4& \hfill 0\\ \hfill 0& \hfill 1& \hfill 2\end{array}\text{}|\text{}\begin{array}{r}\hfill 12\\ \hfill 0\\ \hfill \mathrm{-7}\end{array}\right]$
For the following exercises, solve the system of linear equations using Gaussian elimination.
$\begin{array}{r}3x-4y=-7\\ -6x+8y=14\end{array}$
$\begin{array}{r}3x-4y=1\\ -6x+8y=6\end{array}$
No solutions exist.
$\begin{array}{l}\begin{array}{l}\\ -1.1x-2.3y=6.2\end{array}\hfill \\ -5.2x-4.1y=4.3\hfill \end{array}$
$\begin{array}{r}\hfill 2x+3y+2z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -4x-6y-4z=-2\\ \hfill 10x+15y+10z=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$
No solutions exist.
$\begin{array}{r}\hfill -x+2y-4z=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 3y+8z=-4\\ \hfill -7x+y+2z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$
For the following exercises, find the inverse of the matrix.
$\left[\begin{array}{rr}\hfill -0.2& \hfill 1.4\\ \hfill 1.2& \hfill -0.4\end{array}\right]$
$\frac{1}{8}\left[\begin{array}{cc}2& 7\\ 6& 1\end{array}\right]$
$\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}\\ \hfill -\frac{1}{4}& \hfill \frac{3}{4}\end{array}\right]$
$\left[\begin{array}{ccc}12& 9& -6\\ -1& 3& 2\\ -4& -3& 2\end{array}\right]$
No inverse exists.
$\left[\begin{array}{ccc}2& 1& 3\\ 1& 2& 3\\ 3& 2& 1\end{array}\right]$
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