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In the previous lesson, we examined the derivation of different formulae for solving different types of equations so that we don’t have to go through the process of solving for the unknown variable from first principles. But Vedic Mathematics provides another method of solving some equations on sight that we will examine in this lesson.

You can find all the previous posts about Vedic Mathematics below:

Introduction to Vedic Mathematics
A Spectacular Illustration of Vedic Mathematics
10’s Complements
Multiplication Part 1
Multiplication Part 2
Multiplication Part 3
Multiplication Part 4
Multiplication Part 5
Multiplication Special Case 1
Multiplication Special Case 2
Multiplication Special Case 3
Vertically And Crosswise I
Vertically And Crosswise II
Squaring, Cubing, Etc.
Division By The Nikhilam Method I
Division By The Nikhilam Method II
Division By The Nikhilam Method III
Division By The Paravartya Method
Digital Roots
Straight Division I
Straight Division II
Divisibility Rules
Simple Osculation
Multiplex Osculation
Solving Equations 1

The Vedic method that we are going to examine in this lesson is based on a sutra that reads “Sunyam Samyasamuccaye”. The literal translation of this sutra is that “if the Samuccaya is the same, then that Samuccaya must be zero”.

What exactly does Samuccaya mean though? It turns out that the term means different things under different contexts. This may sound bad at first, but it actually turns out to be a good thing because the more contextual meanings Samuccaya has, the more contexts under which the sutra can be used. And the more contexts under which the sutra can be used, the more types of equations we may be able to solve on sight without expending any labor!

In the full article here, I talk about 5 different meanings of Samuccaya, and some minor extensions, and explain how to use those meanings to solve different types of equations easily. Two of the meanings help us to solve quadratic equations of a special sort without having to expand out the terms, collect them and apply the quadratic formula.

A summary of the results from this lesson can be found in the table below for your reference:

Type Of Equation At-sight solution technique
Unknown quantity is a common factor throughout the equation Set the unknown quantity equal to zero
(x + a)(x + b) = (x + c)(x +d) and ab = cd x = 0
m/(ax + b) + m/(cx + d) = 0 Set the sum of the denominators equal to zero. Thus, x = (–b – d)/(a + c)
Instead of m above, numerator is an unknown quantity In addition to above solution, set the unknown quantity equal to zero for another solution
(ax + b)/(cx + d) = (ex + f)/(gx + h) and ax + b + ex + f = cx + d + gx + h (sum of the numerators = sum of denominators) Equate the sum to zero for one solution
(ax + b)/(cx + d) = (ex + f)/(gx + h) and |ax + b – cx – d| = |ex + f – gx – h| (difference between numerator and denominator is the same on both sides of the equation) Equate the difference to zero for one solution
(ax + b)^2 = (cx + d)^2 Combine the two solution strategies above to derive x = (–b – d)/(a + c) and (if a ≠ c), x = (d – b)/(a – c)

As always, practice is key to success in the application of these methods to the solution of equations. Identifying the samuccaya and applying the right interpretation of the sutra to any given equation takes time and practice. But the rewards can be quite substantial. Good luck, and happy computing!


– The Vedic Maths Forum India