# 9.1 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.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this