Question 118229
{{{1/x + 1/a = 1/b}}} Start with the given equation



{{{1/x  = 1/b - 1/a}}} Subtract {{{1/a}}} from both sides



{{{1/x  = (a/a)(1/b) - (b/b)(1/a)}}} Multiply {{{1/b}}} by {{{a/a}}}. Multiply {{{1/a}}} by {{{b/b}}}.



{{{1/x  = a/ab - b/ab)}}} Multiply.



{{{1/x  = (a- b)/ab}}} Combine the fractions.



{{{ab  = x(a- b)}}} Cross multiply



{{{ab/(a- b) = x}}} Divide both sides by {{{a-b}}}



So our answer is {{{x=ab/(a- b)}}}




note: the previous answer {{{x=-ab/(b- a)}}} is equivalent to {{{x=ab/(a- b)}}}


{{{-ab/(b- a)}}} Start with any answer (I chose the previous answer)



{{{(-1)ab/(b- a)}}} Factor a -1 out of the numerator


{{{(-1)ab/(-1)(-b+ a)}}} Factor a -1 out of the denominator



{{{(-1)ab/(-1)(a-b)}}} Rearrange the terms



{{{cross(-1)ab/cross(-1)(a-b)}}}  Divide and cancel out the "-1" terms



{{{ab/(a-b)}}}  Simplify



So this shows that {{{ab/(a-b)=-ab/(b- a)}}} which means either answer will work.