Fourth Meaning of Samucchaya
Samucchaya also means combination and total. In this context it implies that if the sum of the numerators and the sum of the denominators are equal, then the sum is equated to zero.
Example 1 :
2x + 5 = 2x + 2
2x - 3 2x + 10
N1 + N2 = (2x + 5) + (2x + 2) = 4x + 7
D1 + D2 = (2x - 3) + (2x + 10) = 4x + 7
Hence we find N1 + N2 = D1 + D2
==> 4x + 7 = 0
==> x = -7/4
Example 2 :
3x + 7 = x + 2
6x + 4 2x + 14
Here N1 + N2 = (3x + 7) + (x + 2) = 4x + 9
D1 + D2 = (6x + 4) + (2x + 14) = 8x + 18 = 2(N1 + N2)
In such cases, neglecting the numerical factor, we equate the common term to zero.
==> 4x + 9 = 0
==> x = -9/4
Fifth Meaning of Samucchaya
In the two examples considered above, we find that on cross multiplying the coefficients of x2 on both sides are equal and hence cancel out. But when this is not the case we end up with a quadratic equation.
Samucchaya also includes subtraction and hence we also take into account the difference between numerator and denominator on both sides. i.e. to obtain the first value of x, we check if
N1 + N2 = D1 + D2
To obtain the second value of x, we check if N1 ~ D1 = N2 ~ D2 .
If so, we equate the difference to zero to obtain the second value of x.
Example 1 :
4x + 3 = x + 5
2x + 4 = 3x + 4
N1 + N2 = 5x + 8 = D1 + D2
==> 5x + 8 = 0
==> x = -8/5
N1 ~ D1 = 2x - 1 = N2 ~ D2
Hence 2x -1 = 0
==> x = 1/2
We shall deal with more such problems under the heading 'Quadratic Equations'.