Vedas and Mathematics

Sutra 6: If one is in ratio the other is zero is often used to solve simultaneous linear equations which may involve big numbers. But these equations in special cases can be visually solved because of a certain ratio between the coefficients. Consider the following example:

6x + 7y = 8

19x + 14y = 16

Here the ratio of coefficients of y is same as that of the constant terms. Therefore, the "other" variable is zero, i.e., x = 0. Hence, mentally, the solution of the equations is x = 0 and y = 8/7

(alternatively:

19x + 14y = 16 is equivalent to:

(19/2)x +7y = 8.

Thus it is obvious that x has to be zero, no ratio needed, just divide by 2!

Note that it would not work if both had been "in ratio". For then we have the case of coinciding lines with an infinite number of solutions.:

6x + 7y = 8

12x + 14y = 16

This formula is easily applicable to more general cases with any number of variables. For instance

ax + by + cz = a

bx + cy + az = b

cx + ay + bz = c

which yields x = 1, y = 0, z = 0.

A corollary says solving "by addition and by subtraction." It is applicable in case of simultaneous linear equations where the x- and y-coefficients are interchanged. For instance:

45x − 23y = 113

23x − 45y = 91

By addition: 68x − 68 y = 204 => 68 (x − y) = 204 => x − y = 3.

By subtraction: 22x + 22y = 22 => 22 (x + y) = 22 => x + y = 1.

Again, by addition, we eliminate the y-terms: 2x = 4, so x = 2.

Or, by subtraction, we eliminate the x-terms: -2y = 2, and so y = -1.

The solution set is {2,-1}.

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