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For the following exercises, find the solutions by computing the inverse of the matrix.
$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.3x-0.1y=-10\hfill \\ -0.1x+0.3y=14\hfill \end{array}$
$\left(-20,40\right)$
$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.4x-0.2y=-0.6\hfill \\ -0.1x+0.05y=0.3\hfill \end{array}$
$\begin{array}{r}4x+3y-3z=-4.3\\ 5x-4y-z=-6.1\\ x+z=-0.7\end{array}$
$\left(-1,0.2,0.3\right)$
$\begin{array}{r}\hfill \begin{array}{l}\\ -2x-3y+2z=3\end{array}\\ \hfill -x+2y+4z=-5\\ \hfill -2y+5z=-3\end{array}$
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?
17% oranges, 34% bananas, 39% apples
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?
For the following exercises, find the determinant.
$\left|\begin{array}{cc}0.2& -0.6\\ 0.7& -1.1\end{array}\right|$
$\left|\begin{array}{ccc}-1& 4& 3\\ 0& 2& 3\\ 0& 0& -3\end{array}\right|$
6
$\left|\begin{array}{ccc}\sqrt{2}& 0& 0\\ 0& \sqrt{2}& 0\\ 0& 0& \sqrt{2}\end{array}\right|$
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.
$\begin{array}{r}\hfill 4x-2y=23\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -5x-10y=-35\end{array}$
$\left(6,\frac{1}{2}\right)$
$\begin{array}{l}0.2x-0.1y=0\\ -0.3x+0.3y=2.5\end{array}$
$\begin{array}{r}\hfill -0.5x+0.1y=0.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -0.25x+0.05y=0.15\end{array}$
( x , 5 x + 3)
$\begin{array}{l}x+6y+3z=4\\ 2x+y+2z=3\\ 3x-2y+z=0\end{array}$
$\begin{array}{r}\hfill 4x-3y+5z=-\frac{5}{2}\\ \hfill 7x-9y-3z=\frac{3}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill x-5y-5z=\frac{5}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$
$\left(0,0,-\frac{1}{2}\right)$
$\begin{array}{r}\frac{3}{10}x-\frac{1}{5}y-\frac{3}{10}z=-\frac{1}{50}\\ \frac{1}{10}x-\frac{1}{10}y-\frac{1}{2}z=-\frac{9}{50}\\ \frac{2}{5}x-\frac{1}{2}y-\frac{3}{5}z=-\frac{1}{5}\end{array}$
Is the following ordered pair a solution to the system of equations?
$\begin{array}{l}\\ \begin{array}{l}-5x-y=12\text{\hspace{0.17em}}\hfill \\ x+4y=9\hfill \end{array}\end{array}$ with $\text{\hspace{0.17em}}(-3,3)$
Yes
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
$\begin{array}{r}\frac{1}{2}x-\frac{1}{3}y=4\\ \frac{3}{2}x-y=0\end{array}$
$\begin{array}{r}\hfill \begin{array}{l}\\ -\frac{1}{2}x-4y=4\end{array}\\ \hfill 2x+16y=2\end{array}$
No solutions exist.
$\begin{array}{r}\hfill 5x-y=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -10x+2y=-2\end{array}$
$\begin{array}{l}4x-6y-2z=\frac{1}{10}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-7y+5z=-\frac{1}{4}\hfill \\ 3x+6y-9z=\frac{6}{5}\hfill \end{array}$
$\frac{1}{20}\left(10,5,4\right)$
$\begin{array}{r}x+z=20\\ x+y+z=20\\ x+2y+z=10\end{array}$
$\begin{array}{r}5x-4y-3z=0\\ 2x+y+2z=0\\ x-6y-7z=0\end{array}$
$\left(x,\frac{16x}{5}-\frac{13x}{5}\right)$
$\begin{array}{l}y={x}^{2}+2x-3\\ y=x-1\end{array}$
$\begin{array}{l}{y}^{2}+{x}^{2}=25\\ {y}^{2}-2{x}^{2}=1\end{array}$
$(-2\sqrt{2},-\sqrt{17}),\left(-2\sqrt{2},\sqrt{17}\right),\left(2\sqrt{2},-\sqrt{17}\right),\left(2\sqrt{2},\sqrt{17}\right)$
For the following exercises, graph the following inequalities.
$y<{x}^{2}+9$
$\begin{array}{l}{x}^{2}+{y}^{2}>4\\ y<{x}^{2}+1\end{array}$
For the following exercises, write the partial fraction decomposition.
$\frac{-8x-30}{{x}^{2}+10x+25}$
$\frac{13x+2}{{(3x+1)}^{2}}$
$\frac{5}{3x+1}-\frac{2x+3}{{(3x+1)}^{2}}$
$\frac{{x}^{4}-{x}^{3}+2x-1}{x{({x}^{2}+1)}^{2}}$
For the following exercises, perform the given matrix operations.
$5\left[\begin{array}{cc}4& 9\\ -2& 3\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}-6& 12\\ 4& -8\end{array}\right]$
$\left[\begin{array}{cc}17& 51\\ -8& 11\end{array}\right]$
$\left[\begin{array}{rrr}\hfill 1& \hfill 4& \hfill -7\\ \hfill -2& \hfill 9& \hfill 5\\ \hfill 12& \hfill 0& \hfill -4\end{array}\right]\text{}\left[\begin{array}{cc}3& -4\\ 1& 3\\ 5& 10\end{array}\right]$
${\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill \frac{1}{3}\\ \hfill \frac{1}{4}& \hfill \frac{1}{5}\end{array}\right]}^{-1}$
$\left[\begin{array}{cc}12& -20\\ -15& 30\end{array}\right]$
$\mathrm{det}\left|\begin{array}{cc}0& 0\\ 400& 4\text{,}000\end{array}\right|$
$\mathrm{det}\left|\begin{array}{rrr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}& \hfill 0\\ \hfill -\frac{1}{2}& \hfill 0& \hfill \frac{1}{2}\\ \hfill 0& \hfill \frac{1}{2}& \hfill 0\end{array}\right|$
$-\frac{1}{8}$
If $\text{\hspace{0.17em}}\mathrm{det}(A)=\mathrm{-6},\text{\hspace{0.17em}}$ 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.
$\left[\begin{array}{rrr}\hfill 14& \hfill -2& \hfill 13\\ \hfill -2& \hfill 3& \hfill -6\\ \hfill 1& \hfill -5& \hfill 12\end{array}\text{}|\text{}\begin{array}{r}\hfill 140\\ \hfill -1\\ \hfill 11\end{array}\right]$
Rewrite the augmented matrix as a system of linear equations.
For the following exercises, use Gaussian elimination to solve the systems of equations.
$\begin{array}{r}x-6y=4\\ 2x-12y=0\end{array}$
No solutions exist.
$$\begin{array}{r}\hfill 2x+y+z=-3\\ \hfill x-2y+3z=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill x-y-z=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$$
For the following exercises, use the inverse of a matrix to solve the systems of equations.
$\begin{array}{r}\hfill 4x-5y=-50\\ \hfill -x+2y=80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$
$\left(100,90\right)$
$\begin{array}{r}\hfill \frac{1}{100}x-\frac{3}{100}y+\frac{1}{20}z=-49\\ \hfill \frac{3}{100}x-\frac{7}{100}y-\frac{1}{100}z=13\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill \frac{9}{100}x-\frac{9}{100}y-\frac{9}{100}z=99\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$
For the following exercises, use Cramer’s Rule to solve the systems of equations.
$\begin{array}{l}200x-300y=2\\ 400x+715y=4\end{array}$
$\left(\frac{1}{100},0\right)$
$\begin{array}{l}0.1x+0.1y-0.1z=-1.2\\ 0.1x-0.2y+0.4z=-1.2\\ 0.5x-0.3y+0.8z=-5.9\end{array}$
For the following exercises, solve using a system of linear equations.
A factory producing cell phones has the following cost and revenue functions: $\text{\hspace{0.17em}}C(x)={x}^{2}+75x+2\text{,}688\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}R(x)={x}^{2}+160x.\text{\hspace{0.17em}}$ What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.
32 or more cell phones per day
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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