y-10=a(x+9), ax-y+9a+10=0 has the solution A(-9,10);
y-2=b(x-9), bx-y-9b+2=0 has the solution B(9,2), where a and b are unknown constants (slopes), and a,b≠0, a≠b.
There are multiple lines passing through A and multiple lines passing through B. We don't know their slopes (an and b), but the point to be found is the intersection of those lines. Therefore the intersection point depends on the unknown slopes.
So, subtracting one equation from the other:
(a-b)x+9(a+b)+8=0, (a-b)x=-9(a+b)-8, x=(9a+9b+8)/(b-a).
y=a(x+9)+10=a(9+(9a+9b+8)/(b-a))=a(18b+8)/(b-a),
y=2a(9b+4)/(b-a).
So the ordered pair which satisfies both equations is:
((9a+9b+8)/(b-a),2a(9b+4)/(b-a)).
EXAMPLES
If b=a+1 then this pair becomes: (9a+9a+9+8,2a(9a+9+4)),
(18a+17,18a2+26a).
If a=1, b=2 and the ordered pair is (35,44).