This is, once again, a summary of a longer post authored by me on my own blog. My blog covers a lot of areas, including Vedic Mathematics. If you are interested in reading my thoughts on other topics, please feel free to visit my blog and post comments on the other articles you find there also! Thank you!
In this lesson, we are going to deal with the solution of a special type of equation, using a special type of operation called a merger. During a merger operation, we are going to take several terms of an equation and “merge” them into other terms, leaving us with an equation that is easier to solve (either using basic principles, or using a formula derived from the application of the Paravartya Yojayet sutra).
So, what is the special type of equation that is amenable to solution using the merger operation? Consider the equation below:
1/(x – 4) + 3/(x +2) = 4/(x – 5)
What is special about this equation? Looking at it gives one the impression that it will expand out into a quadratic equation that can be solved using the quadratic formula. However, this equation has a special property that guarantees that it is actually a linear equation whose solution does not involve cross-multiplication, collection of terms and use of the quadratic formula.
That special property is that the sum of numerators on the left hand side of the equation is equal to the numerator on the right hand side of the equation. How is this helpful to our cause? Let us look at the following manipulations of the equation:
1/(x – 4) + 3/(x + 2) = 4/(x – 5) becomes
1/(x – 4) + 3/(x + 2) = 1/(x – 5) + 3/(x – 5) becomes
1/(x – 4) – 1/(x – 5) + 3/(x + 2) – 3/(x – 5) = 0 becomes
[(x – 5) – (x – 4)]/[(x – 4)*(x – 5)] + [3(x – 5) – 3(x + 2)]/[(x + 2)*(x – 5)] = 0 becomes
-1/[(x – 4)*(x – 5)] – 21/[(x + 2)*(x – 5)] = 0 becomes
1/(x – 5) * [-1/(x – 4) – 21/(x + 2)] = 0 becomes
-1/(x – 4) – 21/(x + 2) = 0 becomes
1/(x – 4) + 21/(x + 2) = 0
Thus, the third term of the equation (which was on the right hand side), has been “merged” with the other two terms, giving us an equation with just two terms. Hence the term “merger” for this operation.
Also, we can recognize that the equation we have derived using this merger operation is actually an equation of the 4th type we identified in this earlier lesson on solving equations using the Paravartya Yojayet sutra. Applying the solution formula from that lesson, we can immediately solve the equation and say that x = [-1*2 – 21*(-4)]/(1 + 21), which gives us x = 41/11.
One can verify that the solution derived above is actually the correct solution for the original equation.
So, to formalize the methodology so that we can derive a formula for the solution of equations that can be solved by merger, let us take the general equation below:
p/(x + a) + q/(x + b) = (p + q)/(x + c)
Manipulating this equation using the same steps we used on the example equation above, we get the following series of transformations:
p/(x + a) + q/(x + b) = (p + q)/(x + c) becomes
p/(x + a) + q/(x + b) = p/(x + c) + q/(x + c) becomes
p/(x +a) – p/(x + c) + q/(x + b) – q/(x + c) = 0 becomes
p(x + c – x – a)/[(x + a)*(x + c)] + q(x + c – x – b)/[(x + b)*(x + c)] = 0
Removing the common factor (x + c) from the denominator, we get:
p(c – a)/(x + a) + q(c – b)/(x + b) = 0
We see that the numerators are purely constans, with no unknown terms in them. Therefore, we can now apply the formula for the fourth type of equation from this earlier lesson on deriving formulae for solving linear equations using the Paravartya Yojayet sutra. That gives us:
x = [bp(a – c) + aq(b – c)]/[p(c – a) + q(c – b)]
The formula looks long and complicated when written out using letters, but actually it is easy to apply once we get used to it through a few examples.
For such examples as well as some corollary results that give rise to simpler formulae for the solution provided certain conditions are met, please visit my blog and read the full lesson here. You can also find links to all my previous posts on Vedic Mathematics in the full lesson in my blog.
In this lesson, we have learned about one more simple way of solving linear equations that have a specific structure using an operation called merging. This method allows us to solve such equations mentally and quickly rather than going through the process of expanding out the terms and solving the resulting equation from first principles. In the next lesson, we will look at some more applications of mergers. In the meantime, good luck, and happy computing!