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In the previous lesson, we learned about a powerful application of the Sunyam Samyasamuccaye sutra. We also learned about some ways in which equations could be transformed such that it becomes difficult to identify that the sutra does actually apply to the equation at hand. We then learned how to perform a set of three tests that would reveal whether the given equation can be transformed into an equation of the standard form so that the sutra can then be applied to solve the equation.
In this lesson, we will study another transformation that can hide equations so that it becomes hard to recognize that the sutra is applicable to solve the equation. We will once again start with a simple example.
Consider the equation written below:
4/(4x + 1) + 6/(6x + 3) = 3/(3x + 2) + 12/(12x + 1)
By making the numerator equal to 12 in all the terms (because 12 is the least common multiple of the the current numerators, 3, 4, 6, and 12), we can rewrite the equation as below:
12/(12x + 3) + 12/(12x + 6) = 12/(12x + 8) + 12/(12x + 1)
Now, we recognize that not only are the numerators all identical, but also the sum of the denominators on each side of the equation is 24x + 9. At this point, we recognize that the sutra is applicable, and its application gives us the answer x = -9/24.
Once again, we face the conundrum of whether it would be easier to solve the given equation by expanding the terms or whether we should try the LCM approach without any certainty that the sutra would actually be applicable once we go through the trouble of actually making the numerator the same in all four terms.
It turns out the conundrum is not as bad as it sounds. Once again, there is a test one can perform mentally on the given equation to verify whether it can be transformed into the standard form for application of the sutra.
Consider the given equation to be as below:
a/(bx + c) + d/(ex + f) = g/(hx + j) + k/(mx + n)
To see if the equation can be solved using the samuccaya sutra, calculate the following quantities:
a*(ex + f) + d*(bx + c) and
g*(mx + n) + k*(hx + j)
Obviously, these are the sums of cross-multiplications of the numerators with the denominators of the other term on each side of the equation. If the two sums calculated above are exactly equal to each other, then the second test is passed, and the equation can be transformed into the standard form. Notice that if you take an equation in standard form for which the sutra is applicable, then this test will always be satisfied since bx + c + ex + f = hx + j + mx + n (note that in the standard form, the numerators of the terms are all 1).
Even if the two sums calculated above are not equal to each other, if they are equal to each other after they are reduced to their lowest terms by taking out common multiples, then the test is passed, and the equation can be transformed into the standard form.
If the test is satisfied, then the solution can actually be found even without going through the steps required to convert the given equation into standard form. In fact, once the test is satisfied, the two quantities calculated above become another meaning of Samuccaya. Therefore, one can equate either of the two quantities calculated in the test to zero to find the solution to the given equation.
Let us consider some simple examples to make sure the test and the final solution are fully understood. Take the case of 2/(2x + 3) + 6/(6x + 5) = 3/(3x + 1) + 4/(4x + 8)
To perform the test, we crossmultiply the numerators and denominators on the left hand side, and we get 6*(2x + 3) + 2*(6x + 5) = 24x + 28. On the right hand side, we get 3*(4x + 8) + 4*(3x + 1) = 24x + 28. Since the two quantities are equal, we can tell that the test is passed. We then equate 24x + 28 to zero, getting us the solution x = -7/6.
Let us try it out on a different example now. Consider the equation 2/(2x + 3) – 1/(x + 1) = 6/(6x + 7) – 3/(3x + 2). We calculate the sum of the cross-products on the left hand side of the equation as 2*(x + 1) – 1*(2x + 3) = -1. Similarly, the sum of the cross-products on the right hand side of the equation becomes -2.
Now, both of them are constants, and we can reduce them to the same lowest term, 1. That tells us that the equation can probably be solved using the sutra, but the sums of the cross-products themselves do not lead to the solution since these sums can not be equated to zero.
The first thing we need to recognize about this test is that it is applicable regardless of the order of terms in the equation. As long as there are two terms on either side of the equation, each with a numerator (that is a constant) and a denominator, we can use the sum of cross-products test to verify applicability of the sutra, and then solve the equation if the sutra is applicable.
Therefore, if as above, the sum of the cross-products does not help us given a particular order of terms in the equation, we can try shuffling the terms around and see if things improve. In this case, we notice that all four terms don’t have the same sign. So, one obvious shuffling we can think of is to transpose the negative terms to the opposite sides of the equation, so that we obtain the equation below:
2/(2x + 3) + 3/(3x + 2) = 6/(6x + 7) + 1/(x + 1)
Crossmultiplication and addition gives us 12x + 13 on both sides of the equation. Thus, the test is satisfied once again, and the sum of cross-products came out in a form that is usable for solving the equation. Setting the sum of cross-products to zero, we obtain the solution to the equation as x = -13/12.
The crossmultiplication test in this lesson is very powerful. Its power comes from its ability to identify equations that are amenable to solution using this sutra that don’t have the same numerator in all the terms. In the previous lesson, two of the tests we conducted to see if a given equation could be transformed to the standard form (so that we could attempt to solve it using this sutra) dealt with making sure that the numerators were all the same in all the terms of the equation. That step is actually not needed at all, as we will discover in this lesson.
For full details of how to use the cross-multiplication test to solve equations using the Sunyam Samyasamuccaye sutra, please be sure to read the full lesson here. This will teach you how to combine the tests from the previous lesson with the cross-multiplication test to create a unified methodology that can be applied to lots of equations, enabling one to solve them with much less effort than would be required to solve them using the brute-force method.
The method provided in this lesson is not only simple, but also very powerful in enabling us to ferret out all solutions to a given equation (in case there are more than one) simply by rearranging the terms of the equation! This is another application of this method that is covered in full in the unabridged lesson here.
Once again, practice leads to perfection, so I hope you will take the time to practice the cross-multiplication and other tests for solving equations. Good luck, and happy computing!