Spoiler update solution
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The least common denominator is 6xy.
\(\displaystyle \dfrac{\dfrac{1}{2} \ + \ \dfrac{1}{x} \ - \ \dfrac{1}{y} \ - \dfrac{1}{xy}}{\dfrac{1}{3} \ -\dfrac{1}{x} \ - \dfrac{1}{y} \ + \ \dfrac{1}{xy}} \ \ =\)
\(\displaystyle \dfrac{\bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{2} \ + \ \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{x} \ - \ \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{y} \ - \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{xy}}{ \bigg( \dfrac{6xy}{1} \bigg) \dfrac{1}{3} \ - \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{x} \ - \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{y} \ + \ \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{xy}} \ \ = \)
\(\displaystyle \boxed{ \ \ \dfrac{3xy \ + \ 6y \ - \ 6x \ - 6}{2xy \ - \ 6y \ - \ 6x \ + \ 6} \ \ }\)
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The least common denominator is 6xy.
\(\displaystyle \dfrac{\dfrac{1}{2} \ + \ \dfrac{1}{x} \ - \ \dfrac{1}{y} \ - \dfrac{1}{xy}}{\dfrac{1}{3} \ -\dfrac{1}{x} \ - \dfrac{1}{y} \ + \ \dfrac{1}{xy}} \ \ =\)
\(\displaystyle \dfrac{\bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{2} \ + \ \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{x} \ - \ \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{y} \ - \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{xy}}{ \bigg( \dfrac{6xy}{1} \bigg) \dfrac{1}{3} \ - \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{x} \ - \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{y} \ + \ \bigg(\dfrac{6xy}{1}\bigg)\dfrac{1}{xy}} \ \ = \)
\(\displaystyle \boxed{ \ \ \dfrac{3xy \ + \ 6y \ - \ 6x \ - 6}{2xy \ - \ 6y \ - \ 6x \ + \ 6} \ \ }\)
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