Obtain a diagonal of values with zeros elsewhere to 
find the solution to a system of equations using 
elementary row operations.

Start with this system of equations:

  x +  y +  z =  4
 3x +  y      =  7
       y + 2z = 10

Use elementary row operations to find the solution

R2 <-> R3 
  x +  y +  z =  4
       y + 2z = 10
 3x +  y      =  7

R3 - 3*R1
  x +  y +  z =  4
       y + 2z = 10
     -2y - 3z = -5

R1 - R2
R3 + 2*R2
  x      -  z = -6
       y + 2z = 10
            z = 15

R1 + R3
R2 - 2*R3
  x           =  9
       y +    =-20
            z = 15

Now I'm going to write this without the x, y & z's, 
displaying things in matrix form to the right

  x +  y +  z =  4              1   1   1   4
 3x +  y      =  7              3   1   0   7
       y + 2z = 10              0   1   2  10

Use elementary row operations to find the solution

R2 <-> R3 
  x +  y +  z =  4              1   1   1   4
       y + 2z = 10              0   2   1  10
 3x +  y      =  7              3   1   0   7

R3 - 3*R1
  x +  y +  z =  4              1   1   1   4
       y + 2z = 10              0   1   2  10
     -2y - 3z = -5              0  -2  -3  -5

R1 - R2
R3 + 2*R2
  x      -  z = -6              1   0  -1  -6
       y + 2z = 10              0   1   2  10
            z = 15              0   0   1  15

R1 + R3
R2 - 2*R3
  x           =  9              1   0   0   9
       y +    =-20              0   1   0 -20
            z = 15              0   0   1  15