Sankalana Vyavakalanabhyam

Meaning:

"By addition and by subtraction".

Primary Application:

This sutra provides a very elegant and straightforward method for solving a specific type of system of two simultaneous linear equations. The key characteristic of such a system is that the coefficients of 'x' in the first equation and 'y' in the second equation are the same, AND the coefficients of 'y' in the first equation and 'x' in the second equation are the same. In other words, the coefficients of x and y are interchanged between the two equations.

Format of equations:
ax + by = c₁
bx + ay = c₂

The Method:

1. Addition (Sankalana): Add the two given equations. This will result in a new equation where the coefficients of x and y are (a+b).
   (ax + by) + (bx + ay) = c₁ + c₂
   (a+b)x + (a+b)y = c₁ + c₂
   (a+b)(x+y) = c₁ + c₂
   So, x+y = (c₁ + c₂) / (a+b). Let's call this Equation 3.
2. Subtraction (Vyavakalana): Subtract the second equation from the first (or vice-versa, ensuring consistency).
   (ax + by) - (bx + ay) = c₁ - c₂
   (a-b)x - (a-b)y = c₁ - c₂ (if a-b is taken common, or (a-b)x + (b-a)y )
   (a-b)(x-y) = c₁ - c₂
   So, x-y = (c₁ - c₂) / (a-b). Let's call this Equation 4.
3. Solve the New System: You now have a much simpler system of two linear equations (Equation 3 and Equation 4):
   x + y = P
   x - y = Q
   (where P = (c₁+c₂)/(a+b) and Q = (c₁-c₂)/(a-b))
   Adding these two new equations: 2x = P + Q => x = (P+Q)/2.
   Subtracting the second new equation from the first: 2y = P - Q => y = (P-Q)/2.

Advantages:

This method avoids complex multiplications or divisions often encountered in standard elimination or substitution methods for such specific systems, making calculations quicker and less prone to errors.