Use cramer's rule to solve x_1+2x_2=5 -x_1+x_2=1

Find the linear approximation of the function $f(x)=\sqrt{4-x}$ at $a=0$
Use L(x) to approximate the numbers $\sqrt{3.9}$ and $\sqrt{3.99}$ Round to four decimal places

Use Cramer’s Rule to solve (if possible) the system of linear equations.
2x-y=-10
3x+2y=-1

Fill in the blanks. The process used to write a system of linear equations in row-echelon form is called ________ elimination.

In a previous task, I determined a polynomial interpolation using a system of linear equations.
The data points to be used were
$((0,f(0)),(\frac{1}{6},f(\frac{1}{6})),(\frac{1}{4},f(\frac{1}{4})))$
The linear equation used was of the form:
$((0^2,0^1,0^0),({\frac{1}{6}}^2,{\frac{1}{6}}^1,{\frac{1}{6}}^0),({\frac{1}{4}}^2,{\frac{1}{4}}^1,{\frac{1}{4}}^0))$
$((a_2),(a_1),(a_0))$
$((0),(\frac{1}{\sqrt{3}}),(1))$
and the polynomial of the form
${\displaystyle p_2\left(x\right)=a_0+a_{1}x+a_2{x}^2}$
was determined to have the coefficients
${\displaystyle a_0=0,a_1=-8+\frac{18}{\sqrt{3}},a_2=48-\frac{72}{\sqrt{3}}}$
I am not aware of how I would use a system of linear equations to determine Q(x). Is it perhaps possible to derive the coefficients from p2? Or is there some other method of interpolation I should pursue?

There I have found from an old tutorial notes from a Linear Algebra course I am taking that I am finding the question rather cumbersome. I haven't come across any elementary matrices problem set in this way before. I have tried to using Gauss-Jordan Elimination to reduce matrix A, whilst performing the row operations on the corresponding elementary matrix. Although, I am unsure if this is the correct approach as the problem notes that RREF calculation is not necessary.
If [A|b] denotes the augmented matrix which corresponds to the system of linear equations
${\displaystyle w-y=-2}$
${\displaystyle -w+x+z=5}$
${\displaystyle 2x-y+4z=1}$,
determine the ${\displaystyle 3\times 3}$ matrix E such that [EA|Eb] is the reduced row echelon form of [A|b]. Note: You are asked to calculate the matrix E but you are not asked to calculate the reduced row echelon form.
Any help on how to approach this question or determine the matrix E would be highly appreciated.

I came across the following form of first order linear partial differential equation and I was wondering if there is a name for it?
${\displaystyle \frac{\partial g}{\partial x}\frac{\partial f}{\partial y}-\frac{\partial g}{\partial y}\frac{\partial f}{\partial x}=h(x,y)}$,
where g(x,y) as well as h(x,y) are given and f(x,y) is the unknown.

Find the augmented matrix for the following system of linear equations:
$\{\begin{array}{l}5x+7y-36z=38\\ -8x-11y+57z=-60\end{array}$