Question 412059
<pre>
4x-y+2z=5
  2y+z=4
4x+y+3z=10

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

Erase all the letters

 4 -1 +2 =  5
 0 +2 +1 =  4
 4 +1 +3 = 10

Erase all the + signs

 4 -1  2 =  5
 0  2  1 =  4
 4  1  3 = 10

Replace the equal signs with a |

&#9150;4 -1  2 |  5&#9163;
&#9122;0  2  1 |  4&#9122;
&#9151;4  1  3 | 10&#9164;

We want to end up with a matrix that looks like this:

&#9150;1  0  0 | &#9149;&#9163;
&#9122;0  1  0 | &#9149;&#9122;
&#9151;0  0  1 | &#9149;&#9164;

-------------------------

First we get all the 0's

&#9150;4 -1  2 |  5&#9163;
&#9122;0  2  1 |  4&#9122;
&#9151;4  1  3 | 10&#9164;

To get a 0 where the 4 is on the lower left hand corner,
multiply the top row by -1

&#9150;-4  1 -2 | -5&#9163;
&#9122; 0  2  1 |  4&#9122;
&#9151; 4  1  3 | 10&#9164;

Add the first row to the bottom row:

&#9150;-4  1 -2 | -5&#9163;
&#9122; 0  2  1 |  4&#9122;
&#9151; 0  2  1 |  5&#9164;

To get a 0 where the 1 is in the top row, multiply
the top row by -2

&#9150; 8 -2  4 | 10&#9163;
&#9122; 0  2  1 |  4&#9122;
&#9151; 0  2  1 |  5&#9164;
 
Add the middle row to the top row:

&#9150; 8  0  5 | 14&#9163;
&#9122; 0  2  1 |  4&#9122;
&#9151; 0  2  1 |  5&#9164;

To get a 0 where the 2 on the bottom row is, multiply
the bottom row by -1

&#9150; 8  0  5 | 14&#9163;
&#9122; 0  2  1 |  4&#9122;
&#9151; 0 -2 -1 | -5&#9164;

Add the middle row to the bottom row

&#9150; 8  0  5 | 14&#9163;
&#9122; 0  2  1 |  4&#9122;
&#9151; 0  0  0 | -1&#9164;

If you replace the letters now 

  8x + 0y + 6z = 14
  0x + 2y + 1z =  4
  0x + 0y + 0z =  1

The bottom equation has no solution because the left
side is 0 and the right side is 1, so there can be no
solution.

Edwin</pre>