Solve the following system of equations by elimination method. $\dfrac {2}{x}+\dfrac{2}{3y}=\dfrac{1}{6}, \dfrac{3}{x}+\dfrac{2}{y}=0, x\neq 0, y\neq 0$

Substitute u for 1/x and v for 1/y, changing the system to 2u + (2/3)v equals 1/6 and 3u + 2v equals 0. Multiplying the first equation by 3 yields 6u + 2v equals 1/2, and subtracting the second equation gives 3u equals 1/2, so u equals 1/6. Substituting this into the second equation provides 3(1/6) + 2v equals 0, which means 2v equals -1/2 and v equals -1/4. Since u equals 1/x and v equals 1/y, x equals 6 and y equals -4.

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