Define Absolute system of Linear Equations?

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

1. The graph of a linear equation in two variables is a _____________________line.
2. The general form of linear equation in two variables is _______________________.
3. Linear equations in three variables are expressed as _________________________.
4. The number of variables in is __________________________.
5. The solution of a linear equation always ________________________ the equation.

Define what linear first-order equation forms?

Determine if (1,3) is a solution to the given system of linear equations.
${\displaystyle 5x+y=8}$
${\displaystyle x+2y=5}$

The standard form of a linear firs-order DE is
$\frac{dy}{dx}+P(x)y=Q(x)$
I think the equation is separable if and only if $P(x)$ and $Q(x)$ are constants, but I'm not sure. (Haven't found any counterexamples but also can't seem to prove it.) Can anyone confirm or deny that this is correct?

I know how to use methods like system of linear equations to show that any vector in ${\displaystyle R^n}$ can be expressed in the form of unique linear combinations of a given set of n linearly independent vectors, but how to expand this result to that any given set of exactly n linearly independent vectors must form a basis of ${\displaystyle R^n}$ space? How to understand it conceptually?
I saw there are similar questions, but the answers in those posts do not really convince me.
Thanks in advance.

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