# 11.8 Solving systems with cramer's rule  (Page 9/11)

 Page 9 / 11

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? ## Solving Systems with Cramer's Rule For the following exercises, find the determinant. $|\begin{array}{cc}100& 0\\ 0& 0\end{array}|$ 0 $|\begin{array}{cc}0.2& -0.6\\ 0.7& -1.1\end{array}|$ $|\begin{array}{ccc}-1& 4& 3\\ 0& 2& 3\\ 0& 0& -3\end{array}|$ 6 $|\begin{array}{ccc}\sqrt{2}& 0& 0\\ 0& \sqrt{2}& 0\\ 0& 0& \sqrt{2}\end{array}|$ 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) $\begin{array}{l}x+6y+3z=4\\ 2x+y+2z=3\\ 3x-2y+z=0\end{array}$ $\begin{array}{r}\hfill 4x-3y+5z=-\frac{5}{2}\\ \hfill 7x-9y-3z=\frac{3}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill x-5y-5z=\frac{5}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$ $\left(0,0,-\frac{1}{2}\right)$ $\begin{array}{r}\frac{3}{10}x-\frac{1}{5}y-\frac{3}{10}z=-\frac{1}{50}\\ \frac{1}{10}x-\frac{1}{10}y-\frac{1}{2}z=-\frac{9}{50}\\ \frac{2}{5}x-\frac{1}{2}y-\frac{3}{5}z=-\frac{1}{5}\end{array}$ ## Practice test 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 $\text{\hspace{0.17em}}\left(-3,3\right)$ 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. $\begin{array}{r}\hfill 5x-y=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -10x+2y=-2\end{array}$ $\begin{array}{l}4x-6y-2z=\frac{1}{10}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-7y+5z=-\frac{1}{4}\hfill \\ 3x+6y-9z=\frac{6}{5}\hfill \end{array}$ $\frac{1}{20}\left(10,5,4\right)$ $\begin{array}{r}x+z=20\\ x+y+z=20\\ x+2y+z=10\end{array}$ $\begin{array}{r}5x-4y-3z=0\\ 2x+y+2z=0\\ x-6y-7z=0\end{array}$ $\left(x,\frac{16x}{5}-\frac{13x}{5}\right)$ $\begin{array}{l}y={x}^{2}+2x-3\\ y=x-1\end{array}$ $\begin{array}{l}{y}^{2}+{x}^{2}=25\\ {y}^{2}-2{x}^{2}=1\end{array}$ $\left(-2\sqrt{2},-\sqrt{17}\right),\left(-2\sqrt{2},\sqrt{17}\right),\left(2\sqrt{2},-\sqrt{17}\right),\left(2\sqrt{2},\sqrt{17}\right)$ 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. $\frac{-8x-30}{{x}^{2}+10x+25}$ $\frac{13x+2}{{\left(3x+1\right)}^{2}}$ $\frac{5}{3x+1}-\frac{2x+3}{{\left(3x+1\right)}^{2}}$ $\frac{{x}^{4}-{x}^{3}+2x-1}{x{\left({x}^{2}+1\right)}^{2}}$ For the following exercises, perform the given matrix operations. $5\left[\begin{array}{cc}4& 9\\ -2& 3\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}-6& 12\\ 4& -8\end{array}\right]$ $\left[\begin{array}{cc}17& 51\\ -8& 11\end{array}\right]$ ${\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill \frac{1}{3}\\ \hfill \frac{1}{4}& \hfill \frac{1}{5}\end{array}\right]}^{-1}$ $\left[\begin{array}{cc}12& -20\\ -15& 30\end{array}\right]$ $\mathrm{det}|\begin{array}{cc}0& 0\\ 400& 4\text{,}000\end{array}|$ $\mathrm{det}|\begin{array}{rrr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}& \hfill 0\\ \hfill -\frac{1}{2}& \hfill 0& \hfill \frac{1}{2}\\ \hfill 0& \hfill \frac{1}{2}& \hfill 0\end{array}|$ $-\frac{1}{8}$ If $\text{\hspace{0.17em}}\mathrm{det}\left(A\right)=-6,\text{\hspace{0.17em}}$ 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. $\begin{array}{l}14x-2y+13z=140\hfill \\ -2x+3y-6z=-1\hfill \\ x-5y+12z=11\hfill \end{array}$ Rewrite the augmented matrix as a system of linear equations. $\left[\begin{array}{rrr}\hfill 1& \hfill 0& \hfill 3\\ \hfill -2& \hfill 4& \hfill 9\\ \hfill -6& \hfill 1& \hfill 2\end{array}|\begin{array}{r}\hfill 12\\ \hfill -5\\ \hfill 8\end{array}\right]$ 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: $\text{\hspace{0.17em}}C\left(x\right)={x}^{2}+75x+2\text{,}688\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}R\left(x\right)={x}^{2}+160x.\text{\hspace{0.17em}}$ 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?

#### Questions & Answers

if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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