What is a system of linear equations in three variables?

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

${\displaystyle x-y=-2(1,3)}$
${\displaystyle 3x-y=-2}$
is the ordered pair a solution of the system of linear equations.
or not a solution

Prove that two homogeneous systems of linear equations are equivalent iff they have the same solution set.
My definition of equivalent of linear equations is that each equation of system A is a linear combination of the equations of system B and converse.
So i have a bit trouble proving this statement in general as ${\displaystyle m\times n}$ form. I worked with their coefficient matrices and if i show that these two are row equivalent the problem is solved. Is there any method to use for this? If not what's the general prove.

Please, write the given expression in standard form of linear equations by solving for x.
x – 8 = 0

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.

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.

If a system of linear equations has infinitely many solutions, then the system is called _____. If a system of linear equations has no solution, then the system is called _____.