# 9.8 Solving systems with cramer's rule  (Page 8/11)

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$\left(10,-10,10\right)$

$\begin{array}{r}\hfill 5x+3y-z=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 3x-2y+4z=13\\ \hfill 4x+3y+5z=22\end{array}$

$\begin{array}{r}x+y+z=1\\ 2x+2y+2z=1\\ 3x+3y=2\end{array}$

No solutions exist.

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3x+2y-z=-10\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-y+2z=7\hfill \\ -x+3y+z=-2\hfill \end{array}$

$\left(-1,-2,3\right)$

$\begin{array}{r}\hfill 3x+4z=-11\\ \hfill x-2y=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 4y-z=-10\end{array}$

$\begin{array}{r}2x-3y+z=0\\ 2x+4y-3z=0\\ 6x-2y-z=0\end{array}$

$\left(x,\frac{8x}{5},\frac{14x}{5}\right)$

$\begin{array}{r}6x-4y-2z=2\\ 3x+2y-5z=4\\ 6y-7z=5\end{array}$

For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

11, 17, 33

A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at$15 for students, $12 for children, and$18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

## Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.

$\begin{array}{l}\begin{array}{l}\\ y={x}^{2}-7\end{array}\hfill \\ y=5x-13\hfill \end{array}$

$\left(2,-3\right),\left(3,2\right)$

$\begin{array}{l}\begin{array}{l}\\ y={x}^{2}-4\end{array}\hfill \\ y=5x+10\hfill \end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}=16\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=x-8\hfill \end{array}$

No solution

$\begin{array}{l}{x}^{2}+{y}^{2}=25\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y={x}^{2}+5\hfill \end{array}$

$\begin{array}{r}{x}^{2}+{y}^{2}=4\\ y-{x}^{2}=3\end{array}$

No solution

For the following exercises, graph the inequality.

$y>{x}^{2}-1$

$\frac{1}{4}{x}^{2}+{y}^{2}<4$

For the following exercises, graph the system of inequalities.

$\begin{array}{l}{x}^{2}+{y}^{2}+2x<3\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y>-{x}^{2}-3\hfill \end{array}$

$\begin{array}{l}{x}^{2}-2x+{y}^{2}-4x<4\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y<-x+4\hfill \end{array}$

$\begin{array}{l}{x}^{2}+{y}^{2}<1\hfill \\ \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}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{y}^{2}

## Partial Fractions

For the following exercises, decompose into partial fractions.

$\frac{-2x+6}{{x}^{2}+3x+2}$

$\frac{2}{x+2},\frac{-4}{x+1}$

$\frac{10x+2}{4{x}^{2}+4x+1}$

$\frac{7x+20}{{x}^{2}+10x+25}$

$\frac{7}{x+5},\frac{-15}{{\left(x+5\right)}^{2}}$

$\frac{x-18}{{x}^{2}-12x+36}$

$\frac{-{x}^{2}+36x+70}{{x}^{3}-125}$

$\frac{3}{x-5},\frac{-4x+1}{{x}^{2}+5x+25}$

$\frac{-5{x}^{2}+6x-2}{{x}^{3}+27}$

$\frac{{x}^{3}-4{x}^{2}+3x+11}{{\left({x}^{2}-2\right)}^{2}}$

$\frac{x-4}{\left({x}^{2}-2\right)},\frac{5x+3}{{\left({x}^{2}-2\right)}^{2}}$

$\frac{4{x}^{4}-2{x}^{3}+22{x}^{2}-6x+48}{x{\left({x}^{2}+4\right)}^{2}}$

## Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

$A=\left[\begin{array}{rr}\hfill 4& \hfill -2\\ \hfill 1& \hfill 3\end{array}\right],B=\left[\begin{array}{rrr}\hfill 6& \hfill 7& \hfill -3\\ \hfill 11& \hfill -2& \hfill 4\end{array}\right],C=\left[\begin{array}{r}\hfill \begin{array}{cc}6& 7\\ 11& -2\end{array}\\ \hfill \begin{array}{cc}14& 0\end{array}\end{array}\right],D=\left[\begin{array}{rrr}\hfill 1& \hfill -4& \hfill 9\\ \hfill 10& \hfill 5& \hfill -7\\ \hfill 2& \hfill 8& \hfill 5\end{array}\right],E=\left[\begin{array}{rrr}\hfill 7& \hfill -14& \hfill 3\\ \hfill 2& \hfill -1& \hfill 3\\ \hfill 0& \hfill 1& \hfill 9\end{array}\right]$

$-4A$

$\left[\begin{array}{cc}-16& 8\\ -4& -12\end{array}\right]$

$10D-6E$

$B+C$

undefined; dimensions do not match

$AB$

$BA$

undefined; inner dimensions do not match

$BC$

$CB$

$\left[\begin{array}{ccc}113& 28& 10\\ 44& 81& -41\\ 84& 98& -42\end{array}\right]$

$DE$

$ED$

$\left[\begin{array}{ccc}-127& -74& 176\\ -2& 11& 40\\ 28& 77& 38\end{array}\right]$

$EC$

$CE$

undefined; inner dimensions do not match

${A}^{3}$

## Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

$\begin{array}{l}x-3z=7\\ y+2z=-5\text{\hspace{0.17em}}\end{array}$ with infinite solutions

For the following exercises, write the augmented matrix from the system of linear equations.

$\begin{array}{l}\\ \begin{array}{r}\hfill -2x+2y+z=7\\ \hfill 2x-8y+5z=0\\ \hfill 19x-10y+22z=3\end{array}\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4x+2y-3z=14\hfill \\ -12x+3y+z=100\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9x-6y+2z=31\hfill \end{array}$

$\begin{array}{r}\hfill x+3z=12\text{\hspace{0.17em}}\\ \hfill -x+4y=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill y+2z=-7\end{array}$

For the following exercises, solve the system of linear equations using Gaussian elimination.

$\begin{array}{r}3x-4y=-7\\ -6x+8y=14\end{array}$

$\begin{array}{r}3x-4y=1\\ -6x+8y=6\end{array}$

No solutions exist.

$\begin{array}{l}\begin{array}{l}\\ -1.1x-2.3y=6.2\end{array}\hfill \\ -5.2x-4.1y=4.3\hfill \end{array}$

$\begin{array}{r}\hfill 2x+3y+2z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill -4x-6y-4z=-2\\ \hfill 10x+15y+10z=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

No solutions exist.

$\begin{array}{r}\hfill -x+2y-4z=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \hfill 3y+8z=-4\\ \hfill -7x+y+2z=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}$

## Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.

$\left[\begin{array}{rr}\hfill -0.2& \hfill 1.4\\ \hfill 1.2& \hfill -0.4\end{array}\right]$

$\frac{1}{8}\left[\begin{array}{cc}2& 7\\ 6& 1\end{array}\right]$

$\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}\\ \hfill -\frac{1}{4}& \hfill \frac{3}{4}\end{array}\right]$

$\left[\begin{array}{ccc}12& 9& -6\\ -1& 3& 2\\ -4& -3& 2\end{array}\right]$

No inverse exists.

$\left[\begin{array}{ccc}2& 1& 3\\ 1& 2& 3\\ 3& 2& 1\end{array}\right]$

#### Questions & Answers

what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike