Vedic Maths Techniques in Algebra

1. If one is in ratio, the other one is zero
This formula is often used to solve simple simultaneous 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 the same as that of the constant terms. Therefore, the “other” is zero, i.e., x = 0. Hence, 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, x has to be zero and no ratio is needed, just divide by 2!

Note that it would not work if both had been “in ratio”:
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.
1. When samuccaya is the same, that samuccaya is zero
Consider the following symbols: N1 – Numerator 1, N2 – Numerator 2, D1 – Denominator 1, D2 – Denominator 2 and so on.

This formula is useful for solving equations that can be solved visually. The word “samuccaya” has various meanings in different applications. For instance, it may mean a term, which occurs as a common factor in all the terms concerned.

For example, an equation “12x + 3x = 4x + 5x”. Since “x” occurs as a common factor in all the terms, therefore, x = 0 is the solution.

Alternatively, samuccaya is the product of independent terms. For instance, in (
x + 7) (x + 9) = (x + 3) (x + 21), the samuccaya is 7 × 9 = 3 × 21, therefore, x = 0 is the solution. It is also the sum of the denominators of two fractions having the same numerical numerator, for example:
1/(2x 1) + 1/ (3x  1) = 0 means 5x  2 = 0.

The more commonly used meaning is “combination” or total. For instance, if the sum of the numerators and the sum of denominators are the same then that sum is zero. Therefore,
Therefore, 4x + 16 = 0 or x = 4.

This meaning (“total”) can also be applied in solving the quadratic equations. The total meaning not only imply sum but also subtraction. For instance, when given N1D1 = N2/D2, if N1 + N2 = D1 + D2 (as shown earlier) then this sum is zero. Mental cross multiplication reveals that the resulting equation is quadratic (the coefficients of x2 are different on the two sides). So, if N1  D1 = N2  D2 then that samuccaya is also zero. This yields the other root of a quadratic equation.

The interpretation of “total” is also applied in multi-term RHS and LHS. For instance, consider

Here, D
1 + D2 = D3 + D4 = 2x  16. Thus x = 8.

There are several other cases where samuccaya can be applied with great versatility. For instance, “apparently cubic”
or “biquadratic” equations can be easily solved as shown below:
(x  3)2 + (x  9)3 = 2(x  6)3.

Note that x
3 + x  9 = 2 (x  6). Therefore, (x  6) = 0 or x = 6.

Consider
Observe: N1 + D1 = N2 + D2 = 2x + 8.
Therefore, x = 4.