Question

How do I solve a system of equations using an augmented matrix?

Answer 1

An augmented matrix contains the coefficients of the unknowns and the "pure" coefficients. You can manipulate the rows of this matrix (elementary row operations) to transform the coefficients and to "read", at the end, the solutions of your system.

The two row operations allowed are:
1) swap rows;
2) take the elements of a row, multiply them by a scalar and sum them to the corresponding element of another row.

I show you an example. Let us have the following system:

You can see that now Row 2 contains a zero. This is good because you eliminate one unknown from the equation represented by Row 2 and so you can "see" the value of the remaining one.

You can now apply the same idea to bigger systems with more equations and unknowns.