### Review Exercises

## Systems of Linear Equations: Two Variables

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

$\begin{array}{l}6x-2y=24\\ -3x+3y=18\end{array}$ and $(9,15)$

For the following exercises, use substitution to solve the system of equations.

$\begin{array}{l}\frac{4}{7}x+\frac{1}{5}y=\frac{43}{70}\\ \frac{5}{6}x-\frac{1}{3}y=-\frac{2}{3}\end{array}$

For the following exercises, use addition to solve the system of equations.

$\begin{array}{l}3x+2y=\mathrm{-7}\\ 2x+4y=6\end{array}$

$\begin{array}{l}8x+4y=2\\ 6x-5y=0.7\end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

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

A performer charges $C(x)=50x+10\text{,}000,$ 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?

## Systems of Linear Equations: Three Variables

For the following exercises, solve the system of three equations using substitution or addition.

$\begin{array}{r}\hfill 5x+3y-z=5\\ \hfill 3x-2y+4z=13\\ \hfill 4x+3y+5z=22\end{array}$

$\begin{array}{l}2x-3y+z=\mathrm{-1}\hfill \\ x+y+z=\mathrm{-4}\hfill \\ 4x+2y-3z=33\hfill \end{array}$

$\begin{array}{l}3x+2y-z=\mathrm{-10}\hfill \\ x-y+2z=7\hfill \\ -x+3y+z=\mathrm{-2}\hfill \end{array}$

$\begin{array}{r}\hfill 3x+4z=\mathrm{-11}\\ \hfill x-2y=5\\ \hfill 4y-z=\mathrm{-10}\end{array}$

$\begin{array}{r}6x-4y-2z=2\\ 3x+2y-5z=4\\ 6y-7z=5\end{array}$

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?

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?

## Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.

$\begin{array}{l}y={x}^{2}-4\hfill \\ y=5x+10\hfill \end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}=25\hfill \\ y={x}^{2}+5\hfill \end{array}$

For the following exercises, graph the inequality.

$y>{x}^{2}-1$

For the following exercises, graph the system of inequalities.

$\begin{array}{l}{x}^{2}+{y}^{2}+2x<3\hfill \\ y>-{x}^{2}-3\hfill \end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}<1\hfill \\ {y}^{2}<x\hfill \end{array}$

## Partial Fractions

For the following exercises, decompose into partial fractions.

$\frac{10x+2}{4{x}^{2}+4x+1}$

$\frac{x-18}{{x}^{2}-12x+36}$

$\frac{-5{x}^{2}+6x-2}{{x}^{3}+27}$

$\frac{4{x}^{4}-2{x}^{3}+22{x}^{2}-6x+48}{x{({x}^{2}+4)}^{2}}$

## Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

$10D-6E$

$AB$

$BC$

$DE$

$EC$

${A}^{3}$

## Solving Systems with Gaussian Elimination

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

$\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill \mathrm{-3}\\ \hfill 0& \hfill 1& \hfill 2\\ \hfill 0& \hfill 0& \hfill 0\end{array}|\begin{array}{r}\hfill 7\\ \hfill \mathrm{-5}\\ \hfill 0\end{array}\right]$

$\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 5\\ \hfill 0& \hfill 1& \hfill \mathrm{-2}\\ \hfill 0& \hfill 0& \hfill 0\end{array}|\begin{array}{r}\hfill \mathrm{-9}\\ \hfill 4\\ \hfill 3\end{array}\right]$

For the following exercises, write the augmented matrix from the system of linear equations.

$\begin{array}{l}4x+2y-3z=14\hfill \\ -12x+3y+z=100\hfill \\ 9x-6y+2z=31\hfill \end{array}$

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

$\begin{array}{r}3x-4y=-7\\ -6x+8y=14\end{array}$

$\begin{array}{l}-1.1x-2.3y=6.2\\ -5.2x-4.1y=4.3\hfill \end{array}$

$\begin{array}{r}\hfill -x+2y-4z=8\\ \hfill 3y+8z=-4\\ \hfill -7x+y+2z=1\end{array}$

## Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.

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

$\left[\begin{array}{ccc}2& 1& 3\\ 1& 2& 3\\ 3& 2& 1\end{array}\right]$

For the following exercises, find the solutions by computing the inverse of the matrix.

$\begin{array}{l}0.4x-0.2y=-0.6\hfill \\ -0.1x+0.05y=0.3\hfill \end{array}$

$\begin{array}{r}\hfill \begin{array}{l}\\ -2x-3y+2z=3\end{array}\\ \hfill -x+2y+4z=-5\\ \hfill -2y+5z=-3\end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

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?

A school club 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?

## Solving Systems with Cramer's Rule

For the following exercises, find the determinant.

$\left|\begin{array}{cc}0.2& -0.6\\ 0.7& -1.1\end{array}\right|$

$\left|\begin{array}{ccc}\sqrt{2}& 0& 0\\ 0& \sqrt{2}& 0\\ 0& 0& \sqrt{2}\end{array}\right|$

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

$\begin{array}{l}0.2x-0.1y=0\\ -0.3x+0.3y=2.5\end{array}$

$\begin{array}{l}x+6y+3z=4\\ 2x+y+2z=3\\ 3x-2y+z=0\end{array}$

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

$\begin{array}{r}\frac{3}{10}x-\frac{1}{5}y-\frac{3}{10}z=-\frac{1}{50}\\ \frac{1}{10}x-\frac{1}{10}y-\frac{1}{2}z=-\frac{9}{50}\\ \frac{2}{5}x-\frac{1}{2}y-\frac{3}{5}z=-\frac{1}{5}\end{array}$