Question

How do you solve for `x` in the equation `1/x - 1/a = 1/b`?

Answer 1

`x = (ab)/(a+b)`

Explanation:

We would love to just invert the whole equation to get the answer for `x`. Unfortunately because there are two terms on the LHS, we cannot do that.

`1/x - 1/a = 1/b" isolate the x-term"`

`1/x = 1/b+1/a " add the fractions "`

`1/x = (a+b)/(ab)`

`x = (ab)/(a+b)" invert both sides"`

OR: get rid of the denominators by multiplying each term by the LCD (abx).

`color(red)(abcancelx xx)1/cancelx - color(red)(cancelabx xx)1/cancela = color(red)(acancelbx xx)1/cancelb`

`ab -bx = ax" move x-terms to one side"`

`ab = ax+bx`

`ab = x(a+b)" factorise x"`

`(ab)/(a+b) = x`