Vedic Sutra #6:     If One is in Ratio the Other is Zero

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

Example #1:
    6x + 7y = 8
    19x +14y = 16

  1. Note that the coefficients of y (namely 7 and 14) are in the ratio as are the terms on the right-side (namely 8 and 16)
  2. According to this sutra, we then set "the other" to zero. That means:   x= 0
  3. Hence, from either equation, we get y = - 8/7.
Why?
If (for example) the coefficients of x are in the same ratio as the right-sides (say k).
Then we can rewrite the equations to read:
[1]   Ax +By = C
[2]   kAx + Ey = kC

Then multiply [1] by k and subtract [2] to get::
  (kB-E)y = 0   so "the other" is zero. That is: y = 0.