Write the given expression in slope-intercept form and simplify the

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

Consider a system of linear equations of the form
${\displaystyle \mathbf{A}\mathbf{x}=\mathbf{b},\mathbf{A}\in {\mathbb{R}}^{L\times K},\mathbf{x}\in {\mathbb{R}}^L,\mathbf{b}\in {\mathbb{R}}^K}$
with L variables ${\displaystyle x_1,x_2,\dots ,x_L\in \mathbb{R}}$ and ${\displaystyle K\le L}$ equations.
We are interested in finding a solution for a single variable x_l. Is there an explicit condition for existence of a unique solution for this variable?
Example: if ${\displaystyle x_1+2x_2+3x_3=3}$ and ${\displaystyle 2x_2+3x_3=2}$, then there exist a unique solution ${\displaystyle x_1=1}$ for the variable ${\displaystyle x_1}$, and we cannot find unique solutions for the other variables.

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x,y.x,y. or x,y,z.x,y,z. or $x_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

$\left[\begin{array}{cccccc}1& 0& 0& 0& 1& 0\\ 4& 3& 0& 4& 2& 0\end{array}\right]$

Given:
Rectangular
${\displaystyle l=6x^{2}y}$
${\displaystyle A=72x^3{y}^4}$
Find the width (w)

How would you solve the following system of linear equations:
${\displaystyle 2x_1+(3+a)x_2+2x_3=2+a}$
${\displaystyle x_1+a{x}_2+2x_3=a}$
${\displaystyle a{x}_1+2x_2+2a{x}_3=0}$
assuming that ${\displaystyle a\ne \pm \sqrt{2}}$? I feel confident in solving linear equation systems with just constants as the coefficients but the variable coefficient a is what gives me problems.
Here is the solution I find: solution as augmented matrix but how would the assumption about a make the reduced echelon form any different compared to an assumption about e.g.${\displaystyle a\ne \pm \sqrt{2}}$?

In my understanding, a matrix is all of those:
1. a transformation in a vector space,
2. a function form some domain to some range,
3. a shorthand way of describing a system of linear equations.
By the last point, a real-valued matrix can have any conceivable real numbers as elements. Is that true or does there exist a table of numbers, which cannot be interpreted as a matrix in the linear algebra sense?

Quick way to check if a matrix is diagonalizable.
Is there any quick way to check whether a matrix is diagonalizable or not?
In exam if a question is asked like "Which of the following matrix is diagonalizable?" and four options are given then how can one check it quickly? I hope my question makes sense.