The Sūtra: śūnyaṃ sāmyasamuccaye (If the
samuccaya is same, it is zero) is used to solve equations of six different cases  as explained below. It is normally used by practitioners of Vedic Mathematics, as an additional technique along with conventional methods  to quicken the process for known patterns.
Using Sūtra 5. śūnyaṃ sāmyasamuccaye, for the following case:
ax + bx = cx + dx
If the common factor is same, then the factor is Zero.
As an illustration, let us use this Sūtra for:
7x + 3x = 4x + 5x

Clearly, x = 0, by the Sūtra
By conventional method, the equation gives us:
10x = 9x
or, 10x9x = 0
or, x = 0

Similarly for:
5(x+1) = 3(x+1)

Replacing y = x + 1, we get:
5y = 3y
Clearly, y = 0, by the Sūtra
That is, y = x + 1 = 0
or, x = 1

But, why does it work? For this Sūtra (śūnyaṃ sāmyasamuccaye), let us consider the following:

If ax + bx = cx + dx
Then, x(a + b) = x (c + d)
or, x(a + b  c  d) = 0
This leads to the only possible solution that x = 0, if (a + b  c  d) ≠ 0

Using Sūtra 5. śūnyaṃ sāmyasamuccaye, for the following case:
(x + a)(x + b) = (x + c)(x + d)
If the product of independent terms is same, then variable is Zero.
As an illustration, let us use this Sūtra for:
(x + 3)(x + 4) = (x  2)(x  6)

Product of independent terms:
LHS = 3 × 4 = 12
RHS = 2 × 6 = 12
Clearly, x = 0, by the Sūtra
By conventional method, the equation gives us:
x^{2} + 7x + 12 = x^{2}  8x  12
or, x^{2} + 7x = x^{2}  8x
or, x = 0

But, why does it work? For this Sūtra (śūnyaṃ sāmyasamuccaye), let us consider the following:

If (x + a)(x + b) = (x + c)(x + d)
Then, x^{2} + x(a + b) + ab = x^{2} + x(c + d) + cd
But, ab = cd and thus cancelled out.
or, x^{2} + x(a + b) = x^{2} + x(c + d)
This leads to the only possible solution that x = 0

Using Sūtra 5. śūnyaṃ sāmyasamuccaye, for the following case:
m m
 +  = 0
ax+b cx+d
If the numerators are same, then sum of dinominators is Zero.
As an illustration, let us use this Sūtra for:
5 5
 +  = 0
3x2 2x1

Clearly, (3x2) + (2x1) = 0, by the Sūtra
or, 5x  3 = 0
or, x = 3/5

But, why does it work? For this Sūtra (śūnyaṃ sāmyasamuccaye), let us consider the following:

m m
 +  = 0
ax+b cx+d
Crossmultiplying, we get:
m(cx+d) + m(ax+b) = 0 × (ax+b)(cx+d)
or, m(cx + d + ax + b) = 0
Thus, (cx + d + ax + b) = 0

Using Sūtra 5. śūnyaṃ sāmyasamuccaye, for the following case:
ax+b ax+c
 = 
ax+c ax+b
If the sum of numerators and sum of dinominators is same, then both the sums are Zero.
As an illustration, let us use this Sūtra for:
3x+4 3x+5
 = 
3x+5 3x+4

N_{1} + N_{2} = D_{1} + D_{2} = 6x + 9
Clearly, 6x + 9 = 0, by the Sūtra
or, 6x = 9
or, x = 9/6
or, x = 3/2

But, why does it work? For this Sūtra (śūnyaṃ sāmyasamuccaye), let us consider the following:

ax+b ax+c
 = 
ax+c ax+b
Crossmutiplying, we get:
a^{2}x^{2} + 2abx + b^{2} = a^{2}x^{2} + 2acx + c^{2}
or, 2abx  2acx = c^{2}  b^{2}
or, 2ax (b  c) = (b^{2}  c^{2})
or, 2ax (b  c) =  (b + c)(b  c)
or, 2ax = (b + c)
or, 2ax + (b + c) = 0
Thus, ax + b + ax + c = 0

Using Sūtra 5. śūnyaṃ sāmyasamuccaye, for the following case:
ax+b cx+d
 = 
cx+d ax+b
If the sum of numerators and sum of dinominators is same, and the differences of numerator and dinominator is same, then both the sums are Zero.
As an illustration, let us use this Sūtra for:
3x+2 2x+5
 = 
2x+5 3x+2

N_{1} + N_{2} = D_{1} + D_{2} = 5x + 7
Also, N_{1}  D_{1} = (x  3) = (N_{2}  D_{2})
Clearly, 5x + 7 = 0 and x  3 = 0, by the Sūtra
This gives us:
x = 7/5 or x = 3

But, why does it work? For this Sūtra (śūnyaṃ sāmyasamuccaye), let us consider the following:

ax+b cx+d
 = 
cx+d ax+b
Crossmutiplying, we get:
a^{2}x^{2} + 2abx + b^{2} = c^{2}x^{2} + 2cdx + d^{2}
or, x^{2}(a^{2}  c^{2}) + 2abx  2cdx = d^{2}  b^{2}
Thus, N_{1} + N_{2} = 0 and N_{1}  D_{1} = 0

Using Sūtra 5. śūnyaṃ sāmyasamuccaye, for the following case:
m m m m
 +  =  + 
x+a x+b x+c x+d
If the sum of dinominators is same, then the sum is Zero.
As an illustration, let us use this Sūtra for:
1 1 1 1
 +  =  + 
x+7 x+9 x+6 x+10

D_{1} + D_{2} = D_{3} + D_{4} = 2x + 16
Clearly, 2x + 16 = 0, by the Sūtra
This gives us:
x = 8

Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario  a guideline of which, is presented in
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