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 the previous lesson, we dealt with various meanings of the term Samuccaya, and how the different meanings of the term help us solve different types of equations using the Sunyam Samyasamuccaye sutra.
In this lesson, we will concentrate on one meaning of the term that is very powerful. We will devote this entire lesson to this meaning since there are several variations of this meaning that can take time to identify correctly for application of this sutra.
The basic equation we will start with has the form 1/(x – a) + 1/(x – b) = 1/(x – c) + 1/(x – d). As you can tell, this does not seem to be an easy type of equation to deal with. In fact, at first glance, it looks as if it is likely to become a cubic equation. Actually, if you take the LCM of the denominators, add up the terms on either side, and simplify the equation, you will get an equation of this sort:
[2x – (a + b)][x^2 – (c + d)x + cd] = [2x – (c + d)][x^2 – (a + b)x + ab]
As you can tell, the x^3 terms on the left and right hand side will cancel out, leaving you with a quadratic equation at best, not a cubic equation. But, the prospect of expanding out the terms further, collecting them to create a standard quadratic equation, and then applying the quadratic formula on it to get the final solution is daunting at best.
The meaning of Samuccaya that we are dealing with in this lesson can make problems like this much simpler under some special circumstances. The special circumstance, in this case, happens to be as follows:
If the sum of the denominators on either side of the equation is the same, that is the Samuccaya. Set it equal to zero to solve the equation. Thus, if 2x – (a + b) = 2x – (c + d), then x = (a + b)/2 (or equivalently, x = (c + d)/2) is the solution. That is all there is to it!
Let us see this meaning of Samuccaya in action with a couple of examples. First, let us consider the equation 1/(x + 2) + 1/(x + 3) = 1/(x + 1) + 1/(x + 4). The sum of the denominators on the left hand side is 2x + 5. We find that the sum of the denominators on the right hand side is also 2x + 5. The sutra then says that 2x + 5 = 0, giving us the solution x = -2.5.
Now consider the equation 1/(x – 9) + 1/(x – 7) = 1/(x – 5) + 1/(x – 11). Once again, the sum of denominators on the left hand side is 2x – 16, which is the same as the sum of the denominators on the right hand side. Thus, x = 8.
Note that the numerator need not be 1 in all the terms above. It can be any constant without affecting the outcome. Thus, it is easy to verify that the solution to 3/(x + 2) + 3/(x – 7) = 3/(x – 1) + 3/(x – 4) is also x = 2.5. That is obviously because the equation could be divided by the constant throughout to make all the numerators equal to 1. In fact, the numerators can be any quantity (known or unknown). The sutra is applicable as long as each term has the same numerator.
The reason this meaning can be hard to apply is because there are many ways in which equations can morph themselves so that it is not readily obvious that this meaning is applicable. So, we will now consider some variations of this type of equation that one may encounter. Some of these variations are hard to spot, but once we do, their solution using this meaning of Samuccaya is practically instantaneous. That not only makes this meaning of Samuccaya very powerful, it also makes it very advantageous for us to spot these variations so that we can convert the equations to standard form and get their solution quickly.
If you are interested in the variations of the standard form that I talk about, you can read all about it in the full post here. In the full article, I also talk about how to apply various tests to a given equation so that you can verify that the given equation can actually be transformed to the standard form for application of the sutra.
Obviously, the multitude of ways in which equations can be transformed so that it becomes difficult to identify whether this sutra applies or not, makes it hard to use this meaning of samuccaya in many circumstances. But by being able to perform the tests listed in the lesson mentally and on sight, one can identify equations that are susceptible to this quick and easy solution technique. That can result in large savings of time and effort in actually solving such equations.
Therefore, I hope you will take the time to practice performing the tests on various equations so that you can do them quickly and efficiently. Good luck, and happy computing!