Theory
Move it across, change its sign
When you solve an equation like x + 7 = 10, you 'take the 7 to the other side' and it becomes minus 7, giving x = 10 - 7 = 3. That everyday move has a name in this system: Paraavartya Yojayet, 'transpose and adjust'.
This sutra encodes transposition: moving a term across the equals sign while changing its sign (or its operation). It is the workhorse of solving simple equations, and it also underlies a Vedic method for division. As before, we take it from Bharati Krishna Tirtha's 1965 book and treat it as the technique it describes.
Theory
Transposition, step by step
The rule: a term moved to the other side of the equation has its operation reversed. Something added becomes subtracted; something subtracted becomes added; something multiplying becomes dividing, and vice versa.
Solve 2x + 3 = 11:
1. Transpose the +3: move it across as -3. Now 2x = 11 - 3 = 8.
2. Transpose the multiplying 2: move it across as a division. Now x = 8 / 2 = 4.
Check: 2 times 4 plus 3 is 11. Correct. Each step 'adjusts' by reversing an operation as it crosses, which is exactly what 'transpose and adjust' means.
Practical
Solving ax + b = c by transposition (verified)
def solve_linear(a, b, c):
# ax + b = c -> ax = c - b -> x = (c - b) / a
# 'transpose b (sign change), then transpose a (divide)'
return (c - b) / a
print(solve_linear(1, 7, 10)) # x + 7 = 10 -> 3.0
print(solve_linear(2, 3, 11)) # 2x + 3 = 11 -> 4.0
# Each transposition reverses the operation as the term crosses the '='.This example runs in Gri-Learn on the web, where you can edit it and see the output.
Formula
The same idea also divides
Beyond solving equations, Paraavartya underlies a Vedic method for division, especially by numbers just above a base (like dividing by 11, 12, or 112). The divisor's digits are 'transposed' (their signs flipped) and used to bring down and adjust the answer digit by digit, avoiding long trial division.
We focus here on transposition in equations, because it is the clearest, most useful form of the idea, and the one you will apply most. But keep in mind that 'transpose and adjust' is a single principle with more than one use; consult the prescribed text for the full division method.
Quiz
Solve 2x + 3 = 11 by transposition. What is x?
- x = 7, because 11 minus 3 is 8 and you stop there
- x = 4, because transposing gives 2x = 11 - 3 = 8, then x = 8 / 2 = 4
- x = 28, because you multiply 11 minus 3 by 2
- x = 14, because 2 times 11 minus 3
Show the answer
x = 4, because transposing gives 2x = 11 - 3 = 8, then x = 8 / 2 = 4
Transpose the +3 across the equals sign as -3: 2x = 11 - 3 = 8. Then transpose the multiplying 2 across as a division: x = 8 / 2 = 4. Check: 2 times 4 plus 3 = 11, correct. Option A stops after the first transposition (8 is 2x, not x); you must still divide by 2. Option C multiplies by 2 instead of dividing, reversing the operation the wrong way. Option D mishandles the order and signs entirely. Transposition reverses each operation as its term crosses the equals sign: subtract what was added, then divide by what multiplied.
Think first
Why does a term change sign when it crosses the equals sign?
Transposition feels like a trick. Why is 'move it and flip the sign' actually valid? Then tap.
Show the answer
Because it is a shorthand for doing the SAME operation to BOTH sides of the equation, which keeps the equation balanced. An equation is a statement that two sides are equal, and you may do anything to it as long as you do it to both sides, they stay equal. Take x + 7 = 10. To isolate x, you SUBTRACT 7 from both sides: (x + 7) - 7 = 10 - 7, and on the left the +7 and -7 cancel, leaving x = 10 - 7 = 3. Notice the net effect: the +7 that was on the left has 'appeared' on the right as -7. Transposition is simply skipping the middle step and writing the result directly: a term added on one side moves to the other as subtracted, because subtracting it from both sides is what actually happened. The same logic covers multiplication: 2x = 8 becomes x = 8 / 2 because you DIVIDE both sides by 2, so the multiplying 2 'moves across' as a division. So 'move it and flip' is not a magic trick but a fast, correct bookkeeping of the balance principle, do the same thing to both sides. Understanding this keeps you from misapplying it (for instance, flipping the wrong operation). Balance preserved, shortcut earned.
Summary
Key takeaways
- Paraavartya Yojayet ('transpose and adjust') encodes transposition: move a term across the equals sign and reverse its operation.
- Added becomes subtracted, subtracted becomes added; multiplying becomes dividing, and vice versa.
- Solve 2x + 3 = 11: transpose +3 -> 2x = 8; transpose the 2 -> x = 8 / 2 = 4 (check: 2 times 4 plus 3 = 11).
- In code: x = (c - b) / a solves ax + b = c by the same transpositions.
- The same principle underlies a Vedic division method for numbers near a base (consult the prescribed text).
- Transposition works because it is shorthand for doing the same operation to both sides, keeping the equation balanced.
- Memory hook: cross the equals sign, reverse the operation.