The Sūtra: ānūrupye śūnayamanyat (If one is in ratio, the other is zero) is used to solve simultaneous simple equations where the ratio of coefficients of one of the variables are in ratio of the independent terms.
As an illustration, let us use this Sūtra to solve:
3x + 7y = 2
4x + 21y = 6

Clearly, 7:21 = 1:3
And, 2:6 = 1:3
So, x = 0 (By the Sūtra)
Substituting, we get:
0 + 7y = 2
or, y = 2/7

So, for a practitioner of Vedic Mathematics, for something like:
323x + 147y = 1615
969x + 321y = 4845

Clearly, 323:969 = 1:3
And, 1615:4845 = 1:3
So, y = 0 (By the Sūtra)
Substituting, we get:
323x + 0 = 1615
or, x = 5

Again, for something like:
12x + 78y = 12
16x + 96y = 16

Clearly, 12:16 is the common ratio of the coefficients of x and the independent contants.
So, y = 0 (By the Sūtra)
Substituting, we get:
12x + 0 = 16
or, x = 4/3

This Sūtra also applies for more variables, say:
a_{1}x + b_{1}y + c_{1}z = c_{1}m
a_{2}x + b_{2}y + c_{2}z = c_{2}m
a_{3}x + b_{3}y + c_{3}z = c_{3}m
Then, x = y = 0
But, why does it work? For this Sūtra (ānūrupye śūnayamanyat), let us consider the following:

Consider two equations,
ax + by = bm
cx + dy = dm
Then, dax + dby = dbm
And, bcx + bdy = bdm
Subtracting one from the other, we get:
dax  bcx = 0
x (ad  bc) = 0
Thus, x = 0
Note that, similar proof may be provided for more variables.

Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario  a guideline of which, is presented in
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