# Math Help - Linear Systems and Matrices / Translations of Conics

1. ## Linear Systems and Matrices / Translations of Conics

Solve the system of linear equations and check any solution algebraically.

#20)

5x - 3y + 2z = 3
2x + 4y - z = 7
x - 11y + 4z = 3

Okay so I put this in my T1-84 in matrix a with the proper coefficent values.
I use rref([A]) and this is the solution set I get-

1 0 5/26 0
0 1 -9/26 0
0 0 0 1

Generally the solution is in the last line of solutions. Anyhow, I am very confused on how to solve this by hand to get a solution. Also, is this a special case? A solution and walkthrough would be appreciated.

Write the form or the partial fraction decomposition of the rational rexpression. Do not solve for constants.

#50)

x^2 - 3x + 2
----------------
4x^3 + 11x^2

As long as the highest degree is on the bottom it is not an irrational partial fraction right? I would like the solution and walkthrough on this as well.

Find the center, foci, and vertices of the ellipse, and sketch its graph.

#38)

9x^2 + 4y^2 - 36x + 8y + 31 = 0

I seperate these by terms correct?

(9x^2 - 36x ) + (4y^2 + 8y ) = -31

Do I factor this next?

9(x^2 - 4x) + 4(4^2 + 2y) = -31

If this is the correct track, I am lost from here.

I would like the solution and walkthrough on this as well.

2. $\displaystyle\left\{\begin{array}{lll}5x-3y+2z=3\\2x+4y-z=7\\x-11y+4z=3\end{array}\right.$
Let $\displaystyle A=\begin{pmatrix}5 & -3 & 2\\2 & 4 & -1\\1 & -11 & 4\end{pmatrix}$ be the matrix of system.
We have $\det A=0$.
Let $\displaystyle d=\left|\begin{array}{cc}5 & -3\\2 & 4\end{array}\right|=26$, so the matrix has grade 2.
Let $\displaystyle d_c=\begin{vmatrix}5 & -3 & 3\\2 & 4 & 7\\1 & -11 & 3\end{vmatrix}=364\neq 0\Rightarrow$the system is incompatible.
$\displaystyle \frac{x^2-3x+2}{4x^3+11x^2}=\frac{x^2-3x+2}{x^2(4x+1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C }{4x+11}$