Shunyam Saamyasamuccaye or "When the samuccaya is the same, that samuccaya is zero"

This sutra is useful in solution of several special types of equations that can be solved visually. The word samuccaya has various meanings in different applicatins.

1: It is a term which occurs as a common factor in all the terms concerned

Thus 12x + 3x = 4x + 5x  x is common, hence x = 0

Or 9 (x+1) = 7 (x+1) here (x+1) is common; hence x +1= 0

2: Here Samuccaya means "the product of the independent terms"

Thus, (x +7) (x +9) = (x +3) (x +21)

Here 7 x9 = 3 x 21. Therefore x = 0

3: Samuccaya thirdly means the sum of the Denominators of two fractions having the same numerical numerator

Thus, 1/(2x –1) + 1/(3x –1) = 0 Hence 5x – 2 =0 or x = 2/5

4: Here Samuccaya means combination (or TOTAL).

If the sum of the Numerators and the sum of the Denominators be the same, then that sum = 0

(2x +9)/ (2x +7) = (2x +7)/ (2x +9)

N1 + N2 = D1 + D2 = 2x + 9 + 2x + 7 = 0

Hence 4x + 16 = 0 hence x = -4

Note: If there is a numerical factor in the algebraic sum, that factor should be removed.

(3x +4)/ (6x +7) = (x +1)/ (2x +3)

Here N1 +N2 = 4x +5; D1 +D2 = 8x + 10; 4x +5 =0 x= -5/4

5: Here Samuccaya means TOTAL ie Addition & subtraction

Thus, (3x +4)/ (6x +7) = (5x +6)/ (2x +3)

Here N1+N2 = D1 + D2 = 8x + 10 =0 hence x = - 5/4

D1 – D2 = N2 – N1 = 2x + 2 = 0 x = -1

6: Here Samuccaya means TOTAL; used in Harder equations

Thus, 1/ (x-7) + 1/(x-9) = 1/(x-6) + 1/(x-10)

Vedic Sutra says, (other elements being equal), the sum-total of the denominators on LHS and the total on the RHS are the same, then the total is zero.

Here, D1 + D2 = D3 + D4 = 2x-16 =0 hence x = 8

Examples 1/(x+7) + 1/(x+9) = 1/(x+6) + 1/(x+10) x = - 8

1/(x-7) + 1(x+9) = 1/(x+11) + 1/(x-9) x = - 1

1/(x-8) + 1/(x-9) = 1/(x-5) + 1/(x-12) x = 8-1/2

1/(x-b) - 1/(x-b-d) = 1/(x-c+d) - 1/(x-c) x = 1/2(b+c)

Special Types of seeming Cubics (x- 3)³ + (x –9)³ = 2(x –6)³

current method is very lengthy, but Vedic method says, (x-3) + (x-9) = 2x – 12

Hence x = 6

(x-149)³ + (x-51)³ = 2(x-100)³ Hence 2x-200 =0 & x = 100

(x+a+b-c)³ + (x+b+c-a)³ = 2(x+b)³ x = -b