Ask Uncle Colin: A Fractional Equation

Dear Uncle Colin,

I’m trying to solve $\frac{x}{x-1} = \frac{1}{x-1}$. I think the answer should be 1, but my teacher disagrees. What do you think?

- First Results Are Contradicting Teachers’ - Is One Nonsense?

Hi, FRACTION, and thanks for your message!

It’s tempting, here, to multiply both sides by $(x-1)$ and say “the answer must be 1!” - as I think you’ve done. Unfortunately, there’s a problem: you can only multiply both sides by something that isn’t zero - and when $x=1$, $x-1$ does equal zero.

In fact, neither of the two fractions is defined when $x=1$ - the graph of each expressions has a vertical asymptote there.

Another way to see that it doesn’t work is to bring everything to one side: if you have $\frac{x}{x-1} - \frac{1}{x-1} = 0$, you can make it $\frac{x-1}{x-1}=0$.

The left-hand side of that is 1 everywhere, except when $x=1$, where it’s undefined. That’s never the same as the right hand side, so the equation has no solutions.

Hope that helps!

- Uncle Colin

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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