# सूत्र ७. संकलन व्यवकलनाभ्यां

## (Sūtra 7. saṅkalana vyavakalanābhyāṃ) - By addition, and by subtraction. The Sūtra: saṅkalana vyavakalanābhyāṃ (By addition, and by subtraction) is used to solve simultaneous simple equations which have the co-efficients of the variables interchanged.

As an illustration, let us use this Sūtra, in simple steps, to solve:
3x + 2y = 18
2x + 3y = 17
 Steps Solution 1. Add both the equations, and reduce by common factors. In this case, we get: 5x + 5y = 25, which has 5 as common term. So, we get: x + y = 5 2. Subtract one from the other, and reduce by common factors. In this case, we get: x - y = 1, with no common factor 3. Add and Subtract the two new equations again to get the answer. In this case, adding them again gives: 2x = 6 or, x = 3 And subtracting, we get: 2y = 4 or, y = 2  Thus, we get: x = 3, y = 2, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
45x - 23y = 113
23x - 45y = 91
 Adding, we get: 68x - 68y = 204 or, x - y = 3 Subtracting, we get: 22x + 22y = 22 or, x + y = 1  Adding again, 2x = 4; or, x = 2 Subtracting again, -2y = 2; or, y = -1  Thus, the solution is: x = 2, y = -1

Again, for something like:
1955x - 476y = 2482
476x - 1955y = -4913
 Adding, we get: 2431x - 2431y = -2431 or, x - y = -1 Subtracting, we get: 1479x + 1479y = 7395 or, x + y = 5  Adding again, 2x = 4; or, x = 2 Subtracting again, -2y = -6; or, y = 3  Thus, the solution is: x = 2, y = 3
But, why does it work? For this Sūtra (saṅkalana vyavakalanābhyāṃ), let us consider the following:
 Consider two equations, ax + by = c bx + ay = d  Adding, (a + b)x + (b + a)y = c + d or, x + y = (c + d)/(a + b) Subtracting, (a - b)x + (b - a)y = c - d or, (a - b)x - (a - b)y = c - d or, x - y = (c - d)/(a - b)  Adding again, we get:      (c+d)   (c-d) 2x = ----- + -----      (a+b)   (a-b) Subtracting again, we get:      (c+d)   (c-d) 2y = ----- - -----      (a+b)   (a-b)  Note that, dividing both the equations by 2 provides the solution.

Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »

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