Vedic Sutra #7:     By Addition and by Subtraction

We want to solve for x and y, from:
  Ax + By = C
  Bx + Ay = F

Note: The coefficients of x and y are interchanged !

Example #1:
    45x - 23y = 113
    23x - 45y = 91

  1. Add the two equation and get:     68x - 68y = 204   so   x - y = 3
  2. Subtract the two equations and get:     22x + 22y = 22   so   x + y = 1
  3. Solve for x by adding to get 2x = 4 so x = 2
  4. Solve for y by subtracting to get -2y = -2 so y = -1
Why?
Uh ... there's no need to explain why, eh?
... just add and subtract