# 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

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function