a + b = 3
-b + c = 3
a + 2c = 10
It's so easy after making it into a matrix:
1 1 0 3
0 -1 1 3
1 0 2 10
Where the last column is the augmented matrix (the = part).
Row reduce the rows:
1 0 0 2
0 1 0 1
0 0 1 4
x_1 = 2
x_2 = 1
x_3 = 4
x_1, x_2, x_3 -> a, b, c, respectively.
Try it with the first equation.