What is a system of linear equations? Provide an example with your description.

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

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.

I have some question to ask you that explain to me and remove my confusion.
1.The book said that we call for those equations linear that have first degree power or general forms of linear equations now my question is that we call for those kind of equations linear that have general form of linear equation or can change to general form or other forms of linear equations form example can we call ${\displaystyle \frac{10}{x}=5}$ linear or square root of ${\displaystyle (y-9)=10}$ linear?
2.Can those rational and radical equations that can be manipulated to linear forms called linear equations like the examples above?
3.Are all first degree one variable equations linear?

Solve the system using the elimination method
${\displaystyle x+y=11}$
-x+y=9

Write the correct equation for this line.
${\displaystyle y–y_1=m\left(x–x_1\right)}$, when the slope is 4 and a line passes through the point (-3. 3).

Find the slope of the line and simplify the answer.
-9x = 2

Let A be an abelian group, ${\displaystyle a_i}$ variables indexed with some arbitrary set I and assume we have an infinite set E of linear equations in finitely many variables of the form
${\displaystyle n_1{a}_1+\dots +n_k{a}_k=b}$
with ${\displaystyle n_i\in \mathbb{Z},b\in A}$.
If this system has no solution, then is there already a finite subset of equations in E which are inconsistent? So if any finite subset of equations is solvable, is E solvable?
This is reminiscent of the compactness theorem in 1st order logic, but of course you can't apply this for a concrete group. Am I missing something? I assume that this has an easy proof or a counterexample. What happens if we presume particularly nice groups (divisible, free) or vector spaces? Then this question is just about linear systems of equations.