Technique explanation
The Sankalana-Vyavakalanabhyam sutra ("By addition and by subtraction") provides a rapid method for solving simultaneous linear equations where the coefficients of x and y are swapped between the two equations (e.g., ax + by = m and bx + ay = n).
By adding the two equations, we obtain a simplified equation for x + y. By subtracting them, we obtain a simplified equation for x - y. These two linear equations can then be solved easily to find the values of x and y.
Method Condition: Applies specifically when coefficients are swapped (a₁ = b₂ and b₁ = a₂).
Standard fallback: The method of elimination by equating coefficients or the substitution method.
Worked examples
Example 1: Basic Swapped Coefficients
Solve for x and y: 31x + 23y = 85 and 23x + 31y = 77
- Step 1: Add the equations: (31+23)x + (23+31)y = 85+77 → 54x + 54y = 162 → x + y = 3
- Step 2: Subtract the equations: (31-23)x + (23-31)y = 85-77 → 8x - 8y = 8 → x - y = 1
- Step 3: Solve x + y = 3 and x - y = 1: 2x = 4 → x = 2; 2y = 2 → y = 1
Check: Substitute x=2, y=1 into the second equation: 23(2) + 31(1) = 46 + 31 = 77. The solution is consistent.
Example 2: Large Coefficients
Solve for x and y: 103x + 97y = 303 and 97x + 103y = 297
- Step 1: Add: 200x + 200y = 600 → x + y = 3
- Step 2: Subtract: 6x - 6y = 6 → x - y = 1
- Step 3: Solve: 2x = 4 → x = 2; 2y = 2 → y = 1
Check: Substitute x=2, y=1 into the second equation: 97(2) + 103(1) = 194 + 103 = 297. The solution is consistent.
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