The sutra Shunyam Samyasamuccaye ("When the samuccaya is the same, that samuccaya is zero") provides an instant solution for specific rational equations. In equations of the form (x+a)/(x+b) = (x+c)/(x+d), if the sum of the numerator and denominator is the same on both sides (N1+D1 = N2+D2), that sum can be set to zero to find the root, avoiding complex cross-multiplication.
Standard fallback: Cross-multiplication, resulting in a quadratic equation (which often simplifies to linear if the x² terms cancel).
Example 1: Solve (x + 2)/(x + 3) = (x + 4)/(x + 1)
1. Check Samuccaya: (x + 2) + (x + 3) = 2x + 5. (x + 4) + (x + 1) = 2x + 5. They match.
2. Set to Zero: 2x + 5 = 0.
3. Solve: x = -2.5.
Result: x = -2.5
Check: Substitute x = -2.5 into the original equation: LHS = (-2.5 + 2)/(-2.5 + 3) = -0.5/0.5 = -1. RHS = (-2.5 + 4)/(-2.5 + 1) = 1.5/-1.5 = -1. Since -1 = -1, the solution is correct.
Example 2: Solve (x - 3)/(x + 5) = (x - 1)/(x + 3)
1. Check Samuccaya: (x - 3) + (x + 5) = 2x + 2. (x - 1) + (x + 3) = 2x + 2. They match.
2. Set to Zero: 2x + 2 = 0.
3. Solve: x = -1.
Result: x = -1
Check: Substitute x = -1 into the original equation: LHS = (-1 - 3)/(-1 + 5) = -4/4 = -1. RHS = (-1 - 1)/(-1 + 3) = -2/2 = -1. Since -1 = -1, the solution is correct.