Question 1205181
<pre>
Write the augmented matrix for the given system of equations.

{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
-3, -7,  (p - 6),  2, "|", 1,
2, 0, p^2, p , "|", q^2))}}}

A augmented matrix has no solution if a row 
has all 0's except for a non-zero element, say 1,
for the rightmost element.  For it would
represent the equation 0x+0y+0z+0w=1.

We try to get a row with all 0's except for
a non-zero element for a rightmost element.
Multiply row 1 by 3 and add to row 2:

{{{matrix(3,1,3,1,""^"")}}}{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
-3, -7,  (p - 6),  2, "|", 1,
2, 0, p^2, p , "|", q^2))}}}

{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
0,  2,  p, -1, "|", -2,
2, 0, p^2, p , "|", q^2))}}}

Multiply row 1 by -2 and add to row 3:

{{{matrix(3,1,-2,"",1)}}}{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
-3, -7,  (p - 6),  2, "|", 1,
2, 0, p^2, p , "|", q^2))}}}

{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
0,  2,  p, -1, "|", -2,
0, -6, p^2-4, p+2 , "|", q^2+2))}}}

Multiply row 2 by 3 and add to row 3:

{{{matrix(3,1,"",3,1^"")}}}{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
0,  2,  p, -1, "|", -2,
0, -6, p^2-4, p+2 , "|", q^2+2))}}}

{{{(matrix(3,6,
1,  3,  2, -1, "|", -1,
0,  2,  p, -1, "|", -2,
0,  0, p^2+3p-4, p-1 , "|", q^2-4))}}}

The bottom row will represent an equation with
no solution if:

{{{system(p^2+3p-4=0, p-1=0,q^2-4<>0)}}}

If p=1, the first two equations will be true.
The inequality will hold if {{{q<>"" +- 2}}}

Answer: {{{p=1}}}, {{{q<>"" +- 2}}}

Edwin</pre>