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In the previous lesson, we derived a set of formulae that can be used to solve simultaneous equations in 2 variables. In this lesson we will deal with a couple of special cases that make the solution of some sets of simultaneous equations much simpler than even applying the formulae we derived.

The first special case involves specially structured equations that are easier to solve than by using the application of the formulae we derived in the previous lesson. In particular, in these equations, the coefficients of x and y in both equations are interchanged (along with their signs), leading to the special structure we are about to exploit.

Consider the set of equations below:

10x - 13y = -16

13x - 10y = -7

We immediately see that the coefficients of x and y are interchanged in the two equations along with the signs being changed. One could apply the formulae we derived in the previous lesson to these equations, but that involves multiplications of rather large numbers, which we would rather avoid if we can. It turns out there is a simpler way to solve these equations. The relevant upasutra reads "Sankalana-Vyavakalanabhyam". Literally it means "by addition and by subtraction".

To apply this upasutra, simply add the two equations first. We get:

23x - 23y = -23

Simplify this and we get x - y = -1.

Now, subtract one equation from the other. We get:

-3x - 3y = -9.

Simplifying this leads to x + y = 3. Now, we see that solving these derived equations is much simpler than solving the original given equations. We immediately get the solution x = 1, y = 2 from the two equations we derived by the application of the upasutra.

Let us look at the application of this sutra to one more example. Take the set of equations below:

12x - 23y = 106

23x - 12y = 139

Once again, we are faced with large coefficients that we would rather not multiply and divide with. Applying the sutra, we can derive two equations as below:

35x - 35y = 245 (by addition)

-11x - 11y = -33 (by subtraction)

By eliminating the common factors in both equations, we get:

x - y = 7

x + y = 3

This set is much easier to solve than the original, giving us the solution x = 5, y = -2.

The second special case involves some equations which may look very difficult to solve because they may once again involve large coefficients (at least for one of the unknowns). However, they become very easy to solve because of a vedic sutra that reads "Anurupye Sunyam Anyat". The sutra literally means "if one is in proportion, the other is zero". This is a very powerful sutra that is not limited to simultaneous equations in two variables, but is applicable to any set of simultaneous equations. The application of this powerful sutra to systems of simultaneous equations make their solution much simpler than by brute-force techniques involving the elimination and substitution of variables in the equations.

What exactly does the sutra mean, and how do we apply it to the solution of simultaneous equations? Consider the set of equations below for an explanation:

ax + by = bc

dx + ey = ec

We can see right away that the constants in the above equations are in proportion to the coefficients of y in the equations (b/e = bc/ec). Now, consider the formula we derived for the value of x in the previous lesson. We find that:

x = (bec - bec)/(ae - bd) = 0

This is the practical application of the sutra. Since y is in proportion, "the other", x, is zero according to the sutra. We then see that as soon as x is determined to be zero according to the sutra, the system of equations becomes easy to solve because one can substitute the value of x = 0 in either of the above equations to get the value of y to be c.

Let us look at some applications of this sutra with real equations. First let us consider the system below:

41x + 3y = 63

22x + y = 21

At first glance, this appears to be difficult to solve because the application of our formulae would involve multiplications using large numbers. However, notice that the coefficients of y in the two equations are in proportion to the constants on the right hand sides of the two equations. That is, 63/3 = 21/1. This means that the sutra is applicable, and the application of the sutra immediately tells us that x = 0. The substitution of this value of x in the second equation immediately gives us y = 21. Thus, one can solve the system of equations above almost instantaneously, on sight, by the application of the Sunyam Anyat sutra.

A couple more applications of this methodology are illustrated below:

2x + 21y = 24

3x + 13y = 36

We see that 24/2 = 36/3, so the coefficients of x and the constant are in proportion. By the application of the sutra, therefore, y must be zero. This automatically leads to the solution x = 12.

Similarly, consider:

31x + 14y = 14

16x + 7y = 7

We can immediately see that 14/14 = 7/7. Thus the coefficients of y and the constant are in proportion. Thus, x = 0 and y = 1.

This logic, as mentioned earlier, does not just apply to systems of equations in two variables, but even to larger systems with several more variables. That is what makes this sutra very powerful. But the method has to be applied carefully to avoid errors. For more details and some illustrative examples, read the full post here.

Thus, in this lesson, we have dealt with two ways in which the solution of simultaneous equations can be simplified under certain circumstances. The first set of results relates to systems of two equations in two variables where the coefficients of the two unknowns are found interchanged along with changed signs. We apply the Sankalana-Vyavakalanabhyam upasutra to these cases to simplify the solution process significantly. The second set of results is obtained by the application of the very powerful Anurupye Sunyam Anyat sutra. This sutra is applicable to sets of equations that involve any number of unknowns, and can make the solution of such systems of equations much simpler (if applied carefully and correctly). Learning to identify the applicability of these sutras on sight, and then applying them quickly and correctly requires practice, as always. Hope you take the time to do so! Good luck, and happy computing!!