ax + by + c = 0     (1)
bx + ay + c = 0     (2)


Equivalently

ax + by = -c        (3)
bx + ay = -c        (4)


Multiply eq(3) by b (both sides).  Multiply eq(4) by a (both sides).  You will get


abx + b^2y = -bc    (5)
abx + a^2y = -ac    (6)


Subtract (5) from (6). In this way, you eliminate "x" :


(a^2 - b^2)y = -c*(a-b)  ====>  y =   =  = // cancel (a-b) // = .


What I did is: I started from (3),(4) and eliminated x; in this way I got y.


Now  YOU  return to (3),(4);  eliminate y and get x.