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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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