If we have one equation in three variables, and it is 3x + 4y - 2z = 7 we can rearrange it  to

x = -(4/3) y + (2/3)z + 7/3,

or if we prefer, to

z = 2y + (3/2)x - 7/2

We can change the coefficient matrix from its original form (3, 4, -2 ; 7), with the notation that ; separates left and right side of the equation to any one of (1, 4/3, -2/3; 7/3), (3/4,1, -1/2; 7/4), or (-3/2, -2, 1; -7/2).
If there is a second equation we can subtract an appropriate multiple of our first row to make the coefficient under the 1 into a zero; and so on.