 Proportional Coefficients Method

Sutra : Aanurupye Shoonyam Anyat

Meaning : If one is in ratio, the other one is zero.

We can understand this sutra better by following the given example.

12x - 5y = 21

36x + 7y = 63

Here we find that the ratio of  x coefficients is equal to that of the independent terms. In such cases, according to the sutra, the other variable ( in this case y ) is zero. This yields two simple equations in x which give same value for x.

i.e.          y = 0

12x = 21  ,        36x = 63

=>  x = 21/12 = 7/4     ,    x = 63/36 = 21/12 = 7/4

Some more examples :

1)    6x + 5y =  10

8x + 15y = 30

Here we find     5/15 = 10/30 = 1/3

Hence from the sutra we have x = 0 and

5y = 10    ,       15y = 30

--> y = 2

Hence x = 0 and y = 2.

2)   499x + 172y = 212

9779x + 387y = 477

Here   172/387 = ( 4*43)/( 9*43) = 4/9

212/477 = ( 4*53)/ ( 9*53) = 4/9

Hence from the sutra we have x = 0,

172y = 212           ,           387y = 477

-- >  y = 53/43

NOTE : This rule is also capable of infinite extension and may be extended to any number of unknown quantities.

Example :

Ax + Hy +Jz = A*K

Bx + Dy + Ez = B*K

Cx + Fy + Gz = C*K

(Where K is a constant)

-->  y = 0 ,  z = 0

Ax = A*K   ,     Bx = B*K  ,        Cx = C*K

-->   x = K

Hence  x = K,   y = 0,    z = 0.