Question 1191180
<pre>
{{{(matrix(3,5,
1,1,1,"|",6,
1,2,3,"|",10,
1,2,a,"|",b))}}}

Case 1: a=3.

{{{(matrix(3,5,
1,1,1,"|",6,
1,2,3,"|",10,
1,2,3,"|",10))}}}

{{{(matrix(3,5,
1,1,1,"|",6,
1,2,3,"|",10,
0,0,0,"|",0))}}}

-1R1+R2->R2

{{{(matrix(3,5,
1,1,1,"|",6,
0,1,2,"|",4,
0,0,0,"|",0))}}}

-1R2+R1->R1

{{{(matrix(3,5,
1,0,-1,"|",2,
0,1,2,"|",4,
0,0,0,"|",0))}}}

{{{system(1x+0y-1z=2,0x+1y+2z=4,0x+0y+0z=0)}}}

{{{system(x-z=2,y+2z=4)}}}

{{{system(x=2+z,y=4-2z)}}}

(x,y,z) = (2+z,4-2z,z) where z is arbitrary.

Some solutions from case 1 are (x,y,z)=(2,4,0), (x,y,z)=(3,2,1),
(x,y,z)=(4,0,2), (x,y,z)=(5,-2,3), (x,y,z)=(1,6,-1), etc., etc.,...

Case 2. x ≠ 3    

{{{(matrix(3,5,
1,1,1,"|",6,
1,2,3,"|",10,
1,2,a,"|",b))}}}

-1R2+R1->R1

{{{(matrix(3,5,
1,1,1,"|",6,
1,2,3,"|",10,
1,2,a,"|",b))}}}

-1R1+R2->R2

{{{(matrix(3,5,
1,1,1,"|",6,
0,1,2,"|",4,
1,2,a,"|",b))}}}

-1R1+R3->R3

{{{(matrix(3,5,
1,1,1,"|",6,
0,1,2,"|",4,
0,1,a-1,"|",b-6))}}}

-1R2+R1->R1

{{{(matrix(3,5,
1,0,-1,"|",2,
0,1,2,"|",4,
0,1,a-1,"|",b-6))}}}

-1R2+R3->R1

{{{(matrix(3,5,
1,0,-1,"|",2,
0,1,2,"|",4,
0,1,a-1,"|",b-6))}}}

-1R2+R3->R3

{{{(matrix(3,5,
1,0,-1,"|",2,
0,1,2,"|",4,
0,1,a-3,"|",b-10))}}}

1/(a-3)*R3->R3

{{{(matrix(3,5,
1,0,-1,"|",2,
0,1,2,"|",4,
0,0,1,"|",(b-10)/(a-3)))}}}

R3+R1->R1

{{{(matrix(3,5,
1,0,0,"|",(b-10)/(a-3)+2,
0,1,2,"|",4,
0,0,1,"|",(b-10)/(a-3)))}}}

-2R3+R2->R2

{{{(matrix(3,5,
1,0,0,"|",(b-10)/(a-3)+2,
0,1,0,"|",-2((b-10)/(a-3))+4,
0,0,1,"|",(b-10)/(a-3)))}}}

Simplifying:

{{{(matrix(3,5,
1,0,0,"|",(2a + b - 16)/(a - 3),
0,1,0,"|",(2(2a - b + 4)^"")/(a - 3),
0,0,1,"|",(b-10)/(a-3)))}}}

{{{system(x=(2a + b - 16)/(a - 3),y=(2(2a - b + 4)^"")/(a - 3),z=(b-10)/(a-3))}}}

{{{(matrix(1,5,x,",",y,",",z))}}}{{{""=""}}}{{{(matrix(1,5,(2a + b - 16)/(a - 3),",",(2(2a - b + 4)^"")/(a - 3),",",z=(b-10)/(a-3)))}}}

Edwin</pre>