Vedic Maths Sutra 07 : Saṅkalana-vyavakalanābhyām
Meaning & Essence
This Sūtra means “By addition and by subtraction”.
It is used for solving simultaneous linear equations in which the coefficients of \( x \) and \( y \) are interchanged. By simply adding the two equations and then subtracting them, we obtain a new pair of very simple equations that can be solved instantly.
Mathematical Principle
Consider a system of the special form:
(The coefficients of \( x \) and \( y \) are swapped.)
- Add the two equations → obtain an equation in \( (x + y) \) or \( (x - y) \)
- Subtract the two equations → obtain the other simple equation
- Solve the resulting two simple equations for \( x \) and \( y \)
Worked Examples
- Equations: \( 45x - 23y = 113 \) and \( 23x - 45y = 91 \)
- Add: \( (45x - 23y) + (23x - 45y) = 113 + 91 \)
→ \( 68x - 68y = 204 \) → \( x - y = 3 \) - Subtract: \( (45x - 23y) - (23x - 45y) = 113 - 91 \)
→ \( 22x + 22y = 22 \) → \( x + y = 1 \) - Now solve: \( x - y = 3 \) and \( x + y = 1 \)
Adding → \( 2x = 4 \) → \( x = 2 \)
Subtracting → \( 2y = -2 \) → \( y = -1 \)
- Equations: \( 1955x - 476y = 2482 \) and \( 476x - 1955y = -4913 \)
- Add: \( 2431(x - y) = -2431 \) → \( x - y = -1 \)
- Subtract: \( 1479(x + y) = 7395 \) → \( x + y = 5 \)
- Adding these two results: \( 2x = 4 \) → \( x = 2 \)
- Subtracting: \( 2y = 6 \) → \( y = 3 \)
- Equations: \( 3x + 2y = 18 \) and \( 2x + 3y = 17 \)
- Add: \( 5x + 5y = 35 \) → \( x + y = 7 \)
- Subtract: \( x - y = 1 \)
- Adding: \( 2x = 8 \) → \( x = 4 \)
- Subtracting: \( 2y = 6 \) → \( y = 3 \)
Enter the coefficients for the special form:
a x + b y = c
b x + a y = d
When to Use This Sūtra
• Simultaneous equations in which the coefficients of \( x \) and \( y \) are interchanged
• Systems that look complicated but become trivial after one addition and one subtraction
• Quickly finding \( x + y \) and \( x - y \), then solving for the individual variables
• Excellent for mental calculation when the numbers are large