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$\begin{array}{c}5x-y=4\text{\hspace{0.17em}}\\ x+6y=2\end{array}$ and $\text{\hspace{0.17em}}(4,0)$
$\begin{array}{l}\mathrm{-3}x-5y=13\hfill \\ -x+4y=10\hfill \end{array}$ and $(\mathrm{-6},1)$
Yes
$\begin{array}{c}3x+7y=1\text{\hspace{0.17em}}\\ 2x+4y=0\end{array}$ and $\text{\hspace{0.17em}}(2,3)$
$\begin{array}{l}\mathrm{-2}x+5y=7\hfill \\ \text{}2x+9y=7\hfill \end{array}$ and $(\mathrm{-1},1)$
Yes
$\begin{array}{c}x+8y=43\text{\hspace{0.17em}}\\ 3x\mathrm{-2}y=\mathrm{-1}\end{array}$ and $\text{\hspace{0.17em}}(3,5)$
For the following exercises, solve each system by substitution.
$\begin{array}{l}\text{}x+3y=5\hfill \\ 2x+3y=4\hfill \end{array}$
$(\mathrm{-1},2)$
$\begin{array}{l}\text{}3x\mathrm{-2}y=18\hfill \\ 5x+10y=\mathrm{-10}\hfill \end{array}$
$\begin{array}{l}4x+2y=\mathrm{-10}\\ 3x+9y=0\end{array}$
$(\mathrm{-3},1)$
$\begin{array}{l}2x+4y=\mathrm{-3.8}\\ 9x\mathrm{-5}y=1.3\end{array}$
$\begin{array}{l}\hfill \\ \begin{array}{l}\\ \begin{array}{l}-2x+3y=1.2\hfill \\ -3x-6y=1.8\hfill \end{array}\end{array}\hfill \end{array}$
$\left(-\frac{3}{5},0\right)$
$\begin{array}{l}\text{}x\mathrm{-0.2}y=1\hfill \\ \mathrm{-10}x+2y=5\hfill \end{array}$
$\begin{array}{l}\text{}3x+5y=9\hfill \\ 30x+50y=\mathrm{-90}\hfill \end{array}$
No solutions exist.
$\begin{array}{l}\hfill \\ \begin{array}{l}\text{}\mathrm{-3}x+y=2\hfill \\ 12x\mathrm{-4}y=\mathrm{-8}\hfill \end{array}\hfill \end{array}$
$\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=16\\ \frac{1}{6}x+\frac{1}{4}y=9\end{array}$
$\left(\frac{72}{5},\frac{132}{5}\right)$
$\begin{array}{l}\\ \begin{array}{l}-\frac{1}{4}x+\frac{3}{2}y=11\hfill \\ -\frac{1}{8}x+\frac{1}{3}y=3\hfill \end{array}\end{array}$
For the following exercises, solve each system by addition.
$\begin{array}{l}\hfill \\ \begin{array}{l}\mathrm{-2}x+5y=\mathrm{-42}\hfill \\ \text{}7x+2y=30\hfill \end{array}\hfill \end{array}$
$\left(6,\mathrm{-6}\right)$
$\begin{array}{l}6x\mathrm{-5}y=\mathrm{-34}\\ 2x+6y=4\end{array}$
$\begin{array}{l}\text{}5x-y=\mathrm{-2.6}\hfill \\ \mathrm{-4}x\mathrm{-6}y=1.4\hfill \end{array}$
$\left(-\frac{1}{2},\frac{1}{10}\right)$
$\begin{array}{l}7x\mathrm{-2}y=3\\ 4x+5y=3.25\end{array}$
$\begin{array}{l}\hfill \\ \begin{array}{l}\text{}\mathrm{-x}+2y=\mathrm{-1}\hfill \\ 5x\mathrm{-10}y=6\hfill \end{array}\hfill \end{array}$
No solutions exist.
$\begin{array}{l}\text{}7x+6y=2\hfill \\ \mathrm{-28}x\mathrm{-24}y=\mathrm{-8}\hfill \end{array}$
$\begin{array}{l}\frac{5}{6}x+\frac{1}{4}y=0\\ \frac{1}{8}x-\frac{1}{2}y=-\frac{43}{120}\end{array}$
$\left(-\frac{1}{5},\frac{2}{3}\right)$
$\begin{array}{l}\text{}\frac{1}{3}x+\frac{1}{9}y=\frac{2}{9}\hfill \\ -\frac{1}{2}x+\frac{4}{5}y=-\frac{1}{3}\hfill \end{array}$
$\begin{array}{l}\hfill \\ \begin{array}{l}\mathrm{-0.2}x+0.4y=0.6\hfill \\ \text{}x\mathrm{-2}y=\mathrm{-3}\hfill \end{array}\hfill \end{array}$
$\left(x,\frac{x+3}{2}\right)$
$\begin{array}{l}\begin{array}{l}\\ \mathrm{-0.1}x+0.2y=0.6\end{array}\hfill \\ \text{\hspace{0.17em}}\text{}5x\mathrm{-10}y=1\hfill \end{array}$
For the following exercises, solve each system by any method.
$\begin{array}{l}5x+9y=16\hfill \\ \text{}x+2y=4\hfill \end{array}$
$(\mathrm{-4},4)$
$\begin{array}{l}6x\mathrm{-8}y=\mathrm{-0.6}\\ 3x+2y=0.9\end{array}$
$\begin{array}{l}5x\mathrm{-2}y=2.25\\ 7x\mathrm{-4}y=3\end{array}$
$\left(\frac{1}{2},\frac{1}{8}\right)$
$\begin{array}{l}\begin{array}{l}\hfill \\ \text{}x-\frac{5}{12}y=-\frac{55}{12}\hfill \end{array}\hfill \\ \mathrm{-6}x+\frac{5}{2}y=\frac{55}{2}\hfill \end{array}$
$\begin{array}{l}\\ \begin{array}{l}7x\mathrm{-4}y=\frac{7}{6}\hfill \\ 2x+4y=\frac{1}{3}\hfill \end{array}\end{array}$
$\left(\frac{1}{6},0\right)$
$\begin{array}{l}3x+6y=11\\ 2x+4y=9\end{array}$
$\begin{array}{l}\text{}\frac{7}{3}x-\frac{1}{6}y=2\hfill \\ -\frac{21}{6}x+\frac{3}{12}y=\mathrm{-3}\hfill \end{array}$
$\left(x,2(7x\mathrm{-6})\right)$
$\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=\frac{1}{3}\\ \frac{3}{2}x+\frac{1}{4}y=-\frac{1}{8}\end{array}$
$\begin{array}{l}2.2x+1.3y=\mathrm{-0.1}\\ 4.2x+4.2y=2.1\end{array}$
$\left(-\frac{5}{6},\frac{4}{3}\right)$
$\begin{array}{l}\text{}0.1x+0.2y=2\hfill \\ 0.35x\mathrm{-0.3}y=0\hfill \end{array}$
For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.
$\begin{array}{l}3x-y=0.6\\ x\mathrm{-2}y=1.3\end{array}$
Consistent with one solution
$\begin{array}{l}\begin{array}{l}\\ -x+2y=4\end{array}\hfill \\ \text{}2x\mathrm{-4}y=1\hfill \end{array}$
$\begin{array}{l}\text{}x+2y=7\hfill \\ 2x+6y=12\hfill \end{array}$
Consistent with one solution
$\begin{array}{l}3x\mathrm{-5}y=7\hfill \\ \text{}x\mathrm{-2}y=3\hfill \end{array}$
$\begin{array}{l}\text{}3x\mathrm{-2}y=5\hfill \\ \mathrm{-9}x+6y=\mathrm{-15}\hfill \end{array}$
Dependent with infinitely many solutions
For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.
$\begin{array}{l}\text{}0.1x+0.2y=0.3\hfill \\ \mathrm{-0.3}x+0.5y=1\hfill \end{array}$
$\begin{array}{l}\hfill \\ \begin{array}{l}\mathrm{-0.01}x+0.12y=0.62\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.15x+0.20y=0.52\hfill \end{array}\hfill \end{array}$
$\left(\mathrm{-3.08},4.91\right)$
$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.5x+0.3y=4\hfill \\ 0.25x\mathrm{-0.9}y=0.46\hfill \end{array}$
$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.15x+0.27y=0.39\hfill \\ \mathrm{-0.34}x+0.56y=1.8\hfill \end{array}$
$\left(\mathrm{-1.52},2.29\right)$
$\begin{array}{l}\begin{array}{l}\\ \mathrm{-0.71}x+0.92y=0.13\end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.83x+0.05y=2.1\hfill \end{array}$
For the following exercises, solve each system in terms of $\text{\hspace{0.17em}}A,B,C,D,E,\text{}$ and $\text{\hspace{0.17em}}F\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}A\u2013F\text{\hspace{0.17em}}$ are nonzero numbers. Note that $\text{\hspace{0.17em}}A\ne B\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}AE\ne BD.$
$\begin{array}{l}x+y=A\\ x-y=B\end{array}$
$\left(\frac{A+B}{2},\frac{A-B}{2}\right)$
$\begin{array}{l}x+Ay=1\\ x+By=1\end{array}$
$\begin{array}{l}Ax+y=0\\ Bx+y=1\end{array}$
$\left(\frac{\mathrm{-1}}{A-B},\frac{A}{A-B}\right)$
$\begin{array}{l}Ax+By=C\\ x+y=1\end{array}$
$\begin{array}{l}Ax+By=C\\ Dx+Ey=F\end{array}$
$\left(\frac{CE-BF}{BD-AE},\frac{AF-CD}{BD-AE}\right)$
For the following exercises, solve for the desired quantity.
A stuffed animal business has a total cost of production $\text{\hspace{0.17em}}C=12x+30\text{\hspace{0.17em}}$ and a revenue function $\text{\hspace{0.17em}}R=20x.\text{\hspace{0.17em}}$ Find the break-even point.
A fast-food restaurant has a cost of production $\text{\hspace{0.17em}}C(x)=11x+120\text{\hspace{0.17em}}$ and a revenue function $\text{\hspace{0.17em}}R(x)=5x.\text{\hspace{0.17em}}$ When does the company start to turn a profit?
They never turn a profit.
A cell phone factory has a cost of production $\text{\hspace{0.17em}}C(x)=150x+10,000\text{\hspace{0.17em}}$ and a revenue function $\text{\hspace{0.17em}}R(x)=200x.\text{\hspace{0.17em}}$ What is the break-even point?
A musician charges $\text{\hspace{0.17em}}C(x)=64x+\mathrm{20,000},\text{}$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the total number of attendees at the concert. The venue charges $80 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?
$(1,250,100,000)$
A guitar factory has a cost of production $\text{\hspace{0.17em}}C(x)=75x+\mathrm{50,000.}\text{\hspace{0.17em}}$ If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.
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