Solve the system of equations $\displaystyle\left{\begin{matrix}\dfrac1x+\dfrac1y=\dfrac32\\dfrac1{x^{2}}+\dfrac1{y^{2}}=\dfrac54\end{matrix}\right.$

Multiplying the first equation by 2 gives 2/x + 2/y = 3. Using the algebraic identity that the square of a sum equals the sum of squares plus twice the product, we write (1/x + 1/y)^2 = 1/x^2 + 1/y^2 + 2/(xy). Substituting the known values into the identity yields 9/4 = 5/4 + 2/(xy), which simplifies to 1/(xy) = 1/2. The values 1/x and 1/y are the roots of the quadratic equation t^2 - (3/2)t + 1/2 = 0, which factors to (t - 1)(t - 1/2) = 0 to give roots 1 and 1/2. Solving for the variables gives the solution set as (1, 2).

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