The Sūtra: ānūrupye śūnayamanyat (If one is in ratio, the other is zero) is used to solve simultaneous simple equations where the ratio of co-efficients 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
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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
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So, for a practitioner of Vedic Mathematics, for something like:
323x + 147y = 1615
969x + 321y = 4845
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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
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Again, for something like:
12x + 78y = 12
16x + 96y = 16
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Clearly, 12:16 is the common ratio of the co-efficients of x and the independent contants.
So, y = 0 (By the Sūtra)
Substituting, we get:
12x + 0 = 16
or, x = 4/3
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This Sūtra also applies for more variables, say:
a1x + b1y + c1z = c1m
a2x + b2y + c2z = c2m
a3x + b3y + c3z = c3m
Then, x = y = 0
But, why does it work? For this Sūtra (ānūrupye śūnayamanyat), let us consider the following:
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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.
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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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