### Practice Test

Is the following ordered pair a solution to the system of equations?

$\begin{array}{l}\\ \begin{array}{l}-5x-y=12\hfill \\ x+4y=9\hfill \end{array}\end{array}$ with $(-3,3)$

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

$\begin{array}{r}\frac{1}{2}x-\frac{1}{3}y=4\\ \frac{3}{2}x-y=0\end{array}$

$\begin{array}{r}\hfill \begin{array}{l}\\ -\frac{1}{2}x-4y=4\end{array}\\ \hfill 2x+16y=2\end{array}$

$\begin{array}{r}\hfill 5x-y=1\\ \hfill -10x+2y=-2\end{array}$

$\begin{array}{l}4x-6y-2z=\frac{1}{10}\hfill \\ x-7y+5z=-\frac{1}{4}\hfill \\ 3x+6y-9z=\frac{6}{5}\hfill \end{array}$

$\begin{array}{r}x+z=20\\ x+y+z=20\\ x+2y+z=10\end{array}$

$\begin{array}{l}y={x}^{2}+2x-3\\ y=x-1\end{array}$

For the following exercises, graph the following inequalities.

$y<{x}^{2}+9$

For the following exercises, write the partial fraction decomposition.

$\frac{-8x-30}{{x}^{2}+10x+25}$

$\frac{{x}^{4}-{x}^{3}+2x-1}{x{({x}^{2}+1)}^{2}}$

For the following exercises, perform the given matrix operations.

$5\left[\begin{array}{cc}4& 9\\ -2& 3\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}-6& 12\\ 4& -8\end{array}\right]$

$\left[\begin{array}{rrr}\hfill 1& \hfill 4& \hfill -7\\ \hfill -2& \hfill 9& \hfill 5\\ \hfill 12& \hfill 0& \hfill -4\end{array}\right]\phantom{\rule{0.5em}{0ex}}\left[\begin{array}{cc}3& -4\\ 1& 3\\ 5& 10\end{array}\right]$

${\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill \frac{1}{3}\\ \hfill \frac{1}{4}& \hfill \frac{1}{5}\end{array}\right]}^{-1}$

$\mathrm{det}\left|\begin{array}{cc}0& 0\\ 400& 4\text{,}000\end{array}\right|$

$\mathrm{det}\left|\begin{array}{rrr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}& \hfill 0\\ \hfill -\frac{1}{2}& \hfill 0& \hfill \frac{1}{2}\\ \hfill 0& \hfill \frac{1}{2}& \hfill 0\end{array}\right|$

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

Rewrite the system of linear equations as an augmented matrix.

Rewrite the augmented matrix as a system of linear equations.

For the following exercises, use Gaussian elimination to solve the systems of equations.

$\begin{array}{r}\hfill 2x+y+z=-3\\ \hfill x-2y+3z=6\\ \hfill x-y-z=6\end{array}$

For the following exercises, use the inverse of a matrix to solve the systems of equations.

$\begin{array}{r}\hfill \frac{1}{100}x-\frac{3}{100}y+\frac{1}{20}z=-49\\ \hfill \frac{3}{100}x-\frac{7}{100}y-\frac{1}{100}z=13\\ \hfill \frac{9}{100}x-\frac{9}{100}y-\frac{9}{100}z=99\end{array}$

For the following exercises, use Cramer’s Rule to solve the systems of equations.

$\begin{array}{l}0.1x+0.1y-0.1z=-1.2\\ 0.1x-0.2y+0.4z=-1.2\\ 0.5x-0.3y+0.8z=-5.9\end{array}$

For the following exercises, solve using a system of linear equations.

A factory producing cell phones has the following cost and revenue functions: $C(x)={x}^{2}+75x+2\text{,}688$ 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.

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?