There are a couple of things you have to pay attention to when solving a system of equations.
The first thing you want to pay attention to is the rank of the corresponding matrix, defined as the number of pivot rows in the Reduced Row Echelon form of your matrix (that you get at via Gaussian elimination). You can think of the rank as the number of independent equations. For example, if you have
$a + b = 3$
$2a + 2b = 6$,
those equations are not independent. The second one does not tell you anything that the first one doesn't tell you already.
So instead of characterizing a system as "m equations with n unknowns", treat it as "m independent equations with n unkowns".
The next thing you have to know is how to identify the solution space. Linear algebra tells you that if you have a matrix of rank r and n columns (unkowns), you will have n - r free variables that can take any value. Linear algebra also tells you that the complete solution space consists of any particular solution plus the null space of the matrix. To find both a particular solution and a basis for the null space, you will want to use the Reduced Row Echelon form.