You can find all my previous posts about Vedic Mathematics below:
Introduction to Vedic Mathematics
A Spectacular Illustration of Vedic Mathematics
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
Straight Division I
Straight Division II
Solving Equations 1
Solving Equations 2
Solving Equations 3
Solving Equations 4
Simultaneous Equations 1
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.
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. Consider the set of equations below:
x + y + z = 1
x + 3y + 2z = 2
x + 2y + 3z = 3
We see that 1/1 = 2/2 = 3/3. Thus, the coefficients of z and the constant are in proportion across all the three equations. The sutra actually tells us that the rest of the equation after the removal of the z term in each of the above equations is actually zero, because the "Anyat" in the sutra refers to anything left over after the removal of the terms that are in proportion with the constant.
Now applying the meaning of the sutra and setting the other parts of the equations to zero gives us:
x + y = 0
x + 3y = 0 and
x + 2y = 0
We are left with three equations which can be interpreted by the sutra in two different ways. We could either claim that the coefficients of x and the constant are the in proportion in all equations (and thus set y = 0), or claim that the coefficients of y and the constant are in proportion in all equations (and thus set x = 0). We are then left with the solution x = 0, y = 0, z = 1 for the entire set.
Now consider the set of equations below:
x + y + z = 3
x + y + 2z = 6
x + 3y + 3z = 3
Since the coefficients of z and the constant term are in proportion in the first two equations, we can apply the sutra to these two equations alone, and set x + y = 0 in both these equations. Based on that, we get a solution of z = 3. Substituting this value of z in the third equation gives us x + 3y = -6.
This has now given us two equations in x and y that we can solve easily to get x = 3 and y = -3.
Thus the application of this sutra enables us to solve sets of simultaneous equations involving any number of variables. And the application of this sutra does not stop with individual variables either. Consider the set of equations below:
x + y + z = 3
x + 2y + 2z = 6
x + y - z = 5
We see that the coefficient of (y + z) is in proportion to the value of the constant in the first two equations above. This actually means that the sutra is applicable even though the proportionality is not between the constant terms and the coefficients of just one unknown. The "Anyat" in the sutra refers to whatever is left over after the removal of y + z from the above equations. In this case it is x. Thus, we can set x = 0 in all three equations above. This then gives us the equations:
y + z = 3
2y + 2z = 6
y - z = 5
The second equation is not independent of the first, so we can eliminate it. This then leads us to the solution x = 0, y = 4, z = -1.
What happens if we did not notice that the constant is in proportion to the coefficient of (y + z)? What happens if we thought only that the constant was in proportion to the coefficient of z? We would then get the following set of equations from the first two equations:
x + y = 0
x + 2y = 0
We would also have gotten a solution of z = 3, which when substituted in the third equation gives us:
x + y = 8
Obviously, the first equation and this third equation are not consistent, thus alerting us to the fact that our application of the sutra was not correct. This brings up an interesting point about the application of the sutra: we should be careful to identify the biggest set of unknowns whose coefficients are in proportion to the constants in the given equations. Careless application of the sutra can lead to wrong results (as in almost all walks of life)!
Even though we have stopped the application of the sutra to three variables in this lesson, the sutra has no such in-built limitation. We can apply the sutra to any number of equations in any number of variables. Any time we recognize two equations in which the constants are proportional to any coefficients (or sets of coefficients), we can apply the sutra to simplify the equations, or to derive alternate sets of equations to solve for the final solution. So, this is an extraordinarily powerful sutra that one should always be on the lookout for to recognize applications of, since its application invariably simplifies the process of solving simultaneous equations.
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!!