# 11.1 Systems of linear equations: two variables  (Page 8/20)

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$\begin{array}{c}5x-y=4\text{\hspace{0.17em}}\\ x+6y=2\end{array}$ and $\text{\hspace{0.17em}}\left(4,0\right)$

and $\left(-6,1\right)$

Yes

$\begin{array}{c}3x+7y=1\text{\hspace{0.17em}}\\ 2x+4y=0\end{array}$ and $\text{\hspace{0.17em}}\left(2,3\right)$

and $\left(-1,1\right)$

Yes

$\begin{array}{c}x+8y=43\text{\hspace{0.17em}}\\ 3x-2y=-1\end{array}$ and $\text{\hspace{0.17em}}\left(3,5\right)$

For the following exercises, solve each system by substitution.

$\left(-1,2\right)$

$\begin{array}{l}4x+2y=-10\\ 3x+9y=0\end{array}$

$\left(-3,1\right)$

$\begin{array}{l}2x+4y=-3.8\\ 9x-5y=1.3\end{array}$

$\begin{array}{l}\hfill \\ \begin{array}{l}\\ \begin{array}{l}-2x+3y=1.2\hfill \\ -3x-6y=1.8\hfill \end{array}\end{array}\hfill \end{array}$

$\left(-\frac{3}{5},0\right)$

No solutions exist.

$\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=16\\ \frac{1}{6}x+\frac{1}{4}y=9\end{array}$

$\left(\frac{72}{5},\frac{132}{5}\right)$

$\begin{array}{l}\\ \begin{array}{l}-\frac{1}{4}x+\frac{3}{2}y=11\hfill \\ -\frac{1}{8}x+\frac{1}{3}y=3\hfill \end{array}\end{array}$

For the following exercises, solve each system by addition.

$\left(6,-6\right)$

$\begin{array}{l}6x-5y=-34\\ 2x+6y=4\end{array}$

$\left(-\frac{1}{2},\frac{1}{10}\right)$

$\begin{array}{l}7x-2y=3\\ 4x+5y=3.25\end{array}$

No solutions exist.

$\begin{array}{l}\frac{5}{6}x+\frac{1}{4}y=0\\ \frac{1}{8}x-\frac{1}{2}y=-\frac{43}{120}\end{array}$

$\left(-\frac{1}{5},\frac{2}{3}\right)$

$\left(x,\frac{x+3}{2}\right)$

For the following exercises, solve each system by any method.

$\left(-4,4\right)$

$\begin{array}{l}6x-8y=-0.6\\ 3x+2y=0.9\end{array}$

$\begin{array}{l}5x-2y=2.25\\ 7x-4y=3\end{array}$

$\left(\frac{1}{2},\frac{1}{8}\right)$

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

$\left(\frac{1}{6},0\right)$

$\begin{array}{l}3x+6y=11\\ 2x+4y=9\end{array}$

$\left(x,2\left(7x-6\right)\right)$

$\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=\frac{1}{3}\\ \frac{3}{2}x+\frac{1}{4}y=-\frac{1}{8}\end{array}$

$\begin{array}{l}2.2x+1.3y=-0.1\\ 4.2x+4.2y=2.1\end{array}$

$\left(-\frac{5}{6},\frac{4}{3}\right)$

## Graphical

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.

$\begin{array}{l}3x-y=0.6\\ x-2y=1.3\end{array}$

Consistent with one solution

Consistent with one solution

Dependent with infinitely many solutions

## Technology

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

$\begin{array}{l}\hfill \\ \begin{array}{l}-0.01x+0.12y=0.62\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.15x+0.20y=0.52\hfill \end{array}\hfill \end{array}$

$\left(-3.08,4.91\right)$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.5x+0.3y=4\hfill \\ 0.25x-0.9y=0.46\hfill \end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.15x+0.27y=0.39\hfill \\ -0.34x+0.56y=1.8\hfill \end{array}$

$\left(-1.52,2.29\right)$

$\begin{array}{l}\begin{array}{l}\\ -0.71x+0.92y=0.13\end{array}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0.83x+0.05y=2.1\hfill \end{array}$

## Extensions

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 $\text{\hspace{0.17em}}A–F\text{\hspace{0.17em}}$ are nonzero numbers. Note that $\text{\hspace{0.17em}}A\ne B\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}AE\ne BD.$

$\begin{array}{l}x+y=A\\ x-y=B\end{array}$

$\left(\frac{A+B}{2},\frac{A-B}{2}\right)$

$\begin{array}{l}x+Ay=1\\ x+By=1\end{array}$

$\begin{array}{l}Ax+y=0\\ Bx+y=1\end{array}$

$\left(\frac{-1}{A-B},\frac{A}{A-B}\right)$

$\begin{array}{l}Ax+By=C\\ x+y=1\end{array}$

$\begin{array}{l}Ax+By=C\\ Dx+Ey=F\end{array}$

$\left(\frac{CE-BF}{BD-AE},\frac{AF-CD}{BD-AE}\right)$

## Real-world applications

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 $\text{\hspace{0.17em}}C\left(x\right)=11x+120\text{\hspace{0.17em}}$ and a revenue function $\text{\hspace{0.17em}}R\left(x\right)=5x.\text{\hspace{0.17em}}$ When does the company start to turn a profit?

They never turn a profit.

A cell phone factory has a cost of production $\text{\hspace{0.17em}}C\left(x\right)=150x+10,000\text{\hspace{0.17em}}$ and a revenue function $\text{\hspace{0.17em}}R\left(x\right)=200x.\text{\hspace{0.17em}}$ What is the break-even point?

A musician charges $\text{\hspace{0.17em}}C\left(x\right)=64x+20,000,\text{}$ 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?

$\left(1,250,100,000\right)$

A guitar factory has a cost of production $\text{\hspace{0.17em}}C\left(x\right)=75x+50,000.\text{\hspace{0.17em}}$ 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.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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