your original equations are as follows:
4x − y = 2
8x + 10y = −4 
your matrix would look like the following:

              4      -1       2
              8       10     -4

multiply the first row by 2 and subtract it from the second row to get:

              4       -1        2
              0       12       -8

divide the second row by 12 and add it to the first row to get:

              4        0        16/12
              0       12       -8

multiply the first row by 1/4 and multiply the second row by 1/12 to get:

              1        0        4/12
              0        1        -8/12

your answer should be x = 4/12 and y = -8/12

plug those values into your original equation to get:

4x − y = 2
8x + 10y = −4 

become:
16/12 - (-8/12) = 2
32/12 - 80/12 = -4

these become:
24/12 = 2
-48/12 = -4

these become:
2 = 2
-4 = -4

the original equations are confirmed to be true, establishing the solutions as good.

the gauss-jordan method takes gaussian elimination method one step further by making the elements on the left hand side of the matrix all equal to 1.
this eliminates having to go back and solve for each of the variables in turn.