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When the sum is the same, that sum is zero

Shunyam Saamyasamuccaye

When the Samuccaya (a particular sum or expression) is the same on both sides of an equation, that samuccaya equals zero. The most common application: for 1/(x+a) + 1/(x+b) = 1/(x+c) + 1/(x+d), if a+b = c+d, then x = 0.

How It Works

  1. 1.Given an equation like 1/(x+a) + 1/(x+b) = 1/(x+c) + 1/(x+d).
  2. 2.Compute the sum of the denominators' constants on each side: (a+b) vs (c+d).
  3. 3.If a+b = c+d, this is the 'samuccaya' being equal.
  4. 4.By the sutra, x = 0 is the solution.
  5. 5.Verify by substituting x=0: check 1/a + 1/b = 1/c + 1/d.

Examples

1/(x+2) + 1/(x+8) = 1/(x+4) + 1/(x+6)
Step 1 / 4

Solve: 1/(x+2) + 1/(x+8) = 1/(x+1) + 1/(x+9)

Left1/(x+2) + 1/(x+8)
Right1/(x+1) + 1/(x+9)

Try It Yourself

1/(x+1) + 1/(x+7) = 1/(x+2) + 1/(x+6), x = ?

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