Sankalana-Vyavakalanabhyam: By addition and by subtraction

Sankalana-Vyavakalanabhyam, by addition and by subtraction, solves a pair of simultaneous equations neatly: add them to eliminate one unknown, subtract them to eliminate the other, and read off both answers.

Theory

Two equations, two unknowns

Suppose two numbers add to 10, and their difference is 4. What are they? This is a pair of simultaneous equations, and the sutra Sankalana-Vyavakalanabhyam, 'by addition and by subtraction', solves them with striking neatness.

The idea is elimination: add the two equations to cancel one unknown, and subtract them to cancel the other. Each operation leaves a single simple equation. This lesson works the method with a verified example and codes it, again from Bharati Krishna Tirtha's 1965 book.

Theory

Add to eliminate, subtract to eliminate

Take the system:

x + y = 10

x - y = 4

Add the two equations: the +y and -y cancel, leaving 2x = 14, so x = 7.

Subtract the second from the first: the x terms cancel, leaving 2y = 6, so y = 3.

So x = 7 and y = 3. Check: 7 plus 3 is 10, and 7 minus 3 is 4, both original equations hold. One addition and one subtraction gave both answers, no substitution, no fractions. This is the sutra at its cleanest, when the equations are set up symmetrically.

Practical

Add-and-subtract solver (verified)

def solve_sum_diff(s, d):
    # x + y = s ;  x - y = d
    x = (s + d) / 2   # add the equations:  2x = s + d
    y = (s - d) / 2   # subtract:           2y = s - d
    return x, y

print(solve_sum_diff(10, 4))   # (7.0, 3.0)
# add -> 2x = 14 -> x = 7 ; subtract -> 2y = 6 -> y = 3

This example runs in Gri-Learn on the web, where you can edit it and see the output.

Formula

Why adding and subtracting works

Adding two equations is valid because equals plus equals are equal: if x + y = 10 and x - y = 4, then the two left sides sum to the two right sides, (x + y) + (x - y) = 10 + 4, which simplifies to 2x = 14.

The y term vanishes because +y and -y cancel; that is the whole point of choosing to add here. Subtracting instead cancels the x terms. So you pick add or subtract precisely to make one unknown disappear, leaving a one-variable equation you can solve at once. Same balance principle as transposition, applied to a whole equation.

Quiz

Solve x + y = 10 and x - y = 4 by addition and subtraction. What are x and y?

x = 3, y = 7
x = 7, y = 3, because adding gives 2x = 14 (x = 7) and subtracting gives 2y = 6 (y = 3)
x = 14, y = 6
x = 5, y = 5

Show the answer

x = 7, y = 3, because adding gives 2x = 14 (x = 7) and subtracting gives 2y = 6 (y = 3)

Add the equations: (x + y) + (x - y) = 10 + 4 gives 2x = 14, so x = 7. Subtract the second from the first: (x + y) - (x - y) = 10 - 4 gives 2y = 6, so y = 3. Check: 7 + 3 = 10 and 7 - 3 = 4, both hold. Option A swaps the two values (it would give 3 - 7 = -4, not +4, so it fails the second equation). Option C forgets to halve: 2x = 14 and 2y = 6 give x = 7 and y = 3, not 14 and 6. Option D (5, 5) satisfies the sum but not the difference (5 - 5 = 0, not 4). Add to find x, subtract to find y, then halve.

Think first

When is this add-subtract method especially handy?

General simultaneous equations can be solved by substitution too. When does adding and subtracting shine? Then tap.

Show the answer

It shines when the system is SYMMETRIC, when the two unknowns appear with matching coefficients so that adding or subtracting cleanly cancels one of them, as in the sum-and-difference form x + y = s, x - y = d. In that setup, adding immediately kills y and subtracting immediately kills x, so you get both answers in two effortless steps with no fractions or messy substitution. This pattern is common: 'two numbers whose sum and difference are known', or any pair of equations where one variable has the same coefficient (or equal-and-opposite coefficients) on both. Compared with substitution, which asks you to rearrange one equation, plug it into the other, and simplify, add-and-subtract is faster and less error-prone WHEN the coefficients line up. For messier systems where coefficients do not match, you may first scale an equation to make a coefficient align (which is the general elimination method), or fall back on substitution. So the sutra is not a universal solver; it is the elegant special case of elimination for symmetric systems, exactly the kind that appears often in mental-maths and exam problems. Recognise the sum-and-difference shape, and reach for add-and-subtract. Matching coefficients make elimination effortless.

Summary

Key takeaways

Sankalana-Vyavakalanabhyam ('by addition and by subtraction') solves simultaneous equations by elimination.
Add the two equations to cancel one unknown; subtract them to cancel the other.
For x + y = 10 and x - y = 4: adding gives 2x = 14 (x = 7); subtracting gives 2y = 6 (y = 3).
Check both originals: 7 + 3 = 10 and 7 - 3 = 4.
In code: x = (s + d)/2 and y = (s - d)/2 for the sum-and-difference form.
It is neatest for symmetric systems where adding or subtracting cleanly eliminates a variable; from the 1965 book.
Memory hook: add to eliminate one unknown, subtract to eliminate the other.