Question

How do you solve `3/x = 12/(x+7)`?

Answer 1

The solutions is `x=7/3`.

Explanation:

An identity holds if and only if the identity between the inverses hold (as long as you don't have something like `0=0` of course). So, in you case, you have that

`3/x=12/{x+7} \iff x/3={x+7}/12`

The only thing we have to take care about is to make sure that the original denominators aren't zero, i.e. `x \ne 0` and `x+7 \ne 0`. This means that, when we'll eventually find solutions, we have to make sure that these solutions are nor `0` nor `-7`.

Now let's solve the equality between inverses:

`x/3={x+7}/12 \iff 12x=3(x+7)`.

Expanding, we have `12x=3x+21`. Isolating the `x`-terms and the costants, we get `9x=21`, and thus we can solve for `x`: `x=21/9`, which you can simplify (dividing by `3` both numerator and denominator) into `7/3`.