Abbreviate

3x - 2y = 8
6x - 4y = 1

by the augmented matrix:

[3 -2 | 8]
[6 -4 | 1]

The idea is to use row operations to get 0's
where the 6 and the -2 are, and then to get
1's where the 3 and -4 are.

First let's get a 0 where the 6 is:

Multiply the top row by -2 and the bottom
row by 1:

[-6  4 | -16]
[ 6 -4 |   1]

Now add the top row to the bottom row:

[-6  4 | -16]
[ 0  0 | -15]

Oh oh!  There is no way to get a 0 where 
the 4 is.  So we get a 1 where the -6 is by
dividing the top row through by -6

[-6/-6  4/-6 | -16/-6]
[    0     0 |    -15]

or

[ 1 -2/3 | 8/3]
[ 0    0 | -15]

Now write this matrix as the system of which
it is an abbreviation:

 1x - (2/3)y = 8/3 
 0x +     0y = -15

But the bottom equation says 0 = -15 which
is false and therefore there is no solution.

Such a system is called "inconsistent" which
means if you tried to solve them graphically
you would get two parallel lines which will
never intersect.

Here is the graph of the two lines where you
can see they are parallel:

3x - 2y = 8
6x - 4y = 1



Edwin