*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 + 22x - 3 2x + 10 N_{1}+ N_{2}= (2x + 5) + (2x + 2) = 4x + 7 D_{1}+ D_{2}= (2x - 3) + (2x + 10) = 4x + 7

Hence we find N_{1}+ N_{2}= D_{1}+ D_{2}==> 4x + 7 = 0 ==> x = -7/4

Example 2 :

3x + 7=x + 26x + 4 2x + 14

Here N_{1}+ N_{2}= (3x + 7) + (x + 2) = 4x + 9 D_{1}+ D_{2}= (6x + 4) + (2x + 14) = 8x + 18 = 2(N_{1}+ N_{2})

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 x^{2}
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

N_{1} + N_{2}
= D_{1} + D_{2}

To obtain the second value of x, we check if N_{1}
~ D_{1} = N_{2}
~ D_{2} .

If so, we equate the difference to zero to obtain the second value of x.

Example 1 :

4x + 3=x + 52x + 4 = 3x + 4

N_{1}+ N_{2}= 5x + 8 = D_{1}+ D_{2}==> 5x + 8 = 0 ==> x = -8/5

N_{1}~ D_{1}= 2x - 1 = N_{2}~ D_{2}Hence 2x -1 = 0 ==> x = 1/2

*We shall deal with more such problems under the heading 'Quadratic
Equations'.*