Shunyam Samuccaye

Meaning: "When the Samuccaya is the same, that Samuccaya is zero". 'Samuccaya' is a versatile term that can mean sum, product, combination, or a common factor/expression depending on the context.

Primary Application: This sutra is a powerful tool for solving certain types of algebraic equations where a commonality (the 'Samuccaya') can be identified. If this commonality behaves in specific ways across the equation, it implies that this common factor or expression itself must be equal to zero to satisfy the equation.

Key Scenarios where it's applied:

1. Common Factor in All Terms: If an expression 'X' is a common factor in every term of an equation (e.g., aX + bX = cX or aX + bX = 0), then X = 0 is one of the solutions.
2. Sum of Numerators and Denominators in Fractions:
   • If N1/D1 = N2/D2 and N1 + N2 = D1 + D2 (sum of numerators equals sum of denominators), then this sum is zero, leading to a solution.
   • In an equation like (ax+b)/(px+q) = (cx+d)/(rx+s), if (ax+b) + (cx+d) = (px+q) + (rx+s), then this sum equated to zero gives a solution.
3. If Numerators are Equal: In N/D1 = N/D2, if N is not zero, then D1 = D2. If the equation is N/D1 + N/D2 = 0, then D1 + D2 = 0 gives a solution (assuming N is not zero).
4. If Denominators are Equal: In N1/D = N2/D, then N1 = N2. If N1/D + N2/D = 0, then N1 + N2 = 0 gives a solution (assuming D is not zero).
5. Sum of Numerators is Same on Both Sides, and Sum of Denominators is Same on Both Sides: If N1/D1 + N2/D2 = N3/D3 + N4/D4, and N1+N2 = N3+N4, and D1+D2 = D3+D4, then this condition itself doesn't directly make something zero unless other specific forms are met. The classic application is simpler, e.g., if the sum of numerators of two fractions is equal to the sum of their denominators, then that sum is zero.

A classic "Shunyam Samuccaye" example: If (x+a) is a common factor in an equation, then x+a = 0 (i.e., x = -a) is a solution.