Technique explanation
The sutra Anurupye Shunyamanyat ("If one is in ratio, the other is zero") provides an instant solution for specific systems of simultaneous linear equations. If the ratio of the coefficients of one variable matches the ratio of the constant terms, then the other variable must be zero. This reduces the system to a single-variable equation that can be solved by simple division.
Standard fallback: The method of elimination or substitution to solve the system of simultaneous equations.
Worked examples
Example 1: Solve 6x + 7y = 8 and 19x + 14y = 16
1. Check Ratios: y-coefficients are 7:14 (1:2). Constants are 8:16 (1:2).
2. Apply Sutra: Since the y-ratio matches the constant ratio, x = 0.
3. Solve for y: 7y = 8, so y = 8/7.
Result: x = 0, y = 8/7
Check: Substitute x = 0 and y = 8/7 into the second equation: 19(0) + 14(8/7) = 0 + 2 × 8 = 16. Match confirmed.
Example 2: Solve 12x + 78y = 12 and 16x + 96y = 16
1. Check Ratios: x-coefficients are 12:16 (3:4). Constants are 12:16 (3:4).
2. Apply Sutra: Since the x-ratio matches the constant ratio, y = 0.
3. Solve for x: 12x = 12, so x = 1.
Result: x = 1, y = 0
Check: Substitute x = 1 and y = 0 into the second equation: 16(1) + 96(0) = 16 + 0 = 16. Match confirmed.
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