2250.21 – Strange Equation

There is something very strange about the following equation. What is it?

5+9x−557−x=4x−2015−x5 + \dfrac{9x - 55}{7 - x} = \dfrac{4x - 20}{15 - x}

Solution

OK, we just plug in the algebra machine and see where things go.

We work on the left half of the equation first.

5+9x−557−x=35−5x+9x−557−x=4x−207−x\begin{align*}5 + \dfrac{9x - 55}{7 - x} &= \dfrac{35 - 5x + 9x - 55}{7 - x} \\[1em]&= \dfrac{4 x - 20}{7 - x}\end{align*}

So that's the left hand side. But look: the right hand side is 4x−2015−x\dfrac{4x - 20}{15 - x}!

The only way these two expressions can be equal is if the numerators, 4x−204x - 20, are zero, i.e. if x=5x = 5.

Plug 5 into the original equation to check: Does 5+45−557−5=20−2015−55 + \dfrac{45 - 55}{7 - 5} = \dfrac{20 - 20}{15 - 5}? Yup: 5+−102=05 + \dfrac{-10}{2} = 0