Discussion: Given a system of inhomogeneous equations(equations having a non-zero constant term), here are four examples:

A standard problem is to solve them; which means to reduce them to explicit expressions for x, y and z.
This is feasible in example A; but not in any of the others, for the simple reason that there is not a unique solution for the others.

With no equations, x, y and z can be chosen freely to represent any point in three dimensional space. One linear equation can permit you to solve for one of the three variables in terms of the other two. The space of possible solutions is reduced one dimension by one equation, to a plane.  A second equation can allow you to solve for a second variable in terms of the third, reducing the solution space to a line. With two equations that third variable can still be chosen freely, upon which the other two are determined; there is not one unique solution to the two equations, but at best a whole line of them.

In general, if there are n variables, you need n linear equations to determine them all uniquely; but not any n equations will do.
In example B we have 3 variables and only two equations; obviously not enough to determine a solution.
In cases C and D we have two equations in two variables, but they have the same content in C and contradictory content in D. There is essentially only one equation in case C; and case D is self contradictory.

This leads to the following three basic questions            

1. When are n inhomogeneous linear equations in n variables uniquely solvable?

2. How do you go about solving them?

3. What can you do when they are not uniquely solvable?

Unique Solvability

Discussion:
A single linear equation, a x + b y + c z = d, can be written as rv = d where r is the vector (x, y, z) and v is (a, b, c).
Given v, this equation determines the component of r in the direction of v.
A second equation, say, rw = e, determines the component of r in the direction of w.
As long as v and w have different directions this is new information. The resulting information is enough, by linearity of the dot product, to determine the component of r in any direction that lies in the plane of v and w. (Thus if q = 5 v + 3 w, then rq = 5 rv + 3 rw).
In order that a third vector,t determine new information, it is necessary (and sufficient). that vw and t not lie in the same plane. This is exactly the condition that the volume of the parallelopiped with edges t, v and w is not zero; this is the condition that the determinant formed by taking their components as the rows, does not vanish, since the magnitude of the determinant is this volume. The same reasoning and the same conclusion apply in any higher dimensional space.
Form the square array defined by the coefficients of x, y and z in the three equations: (or in general n equations in n variables). If the determinant of this array or matrix is non-zero, the equations determine a unique solution.

Example