USA Mathematical Talent Search Competition Question | Solve one equation for two variables | Algebra

Given one equation with two variables, how to solve the equation for x and y? This is a question from USA Mathematical Talent Search (USAMTS) competition. In the original problem, there is an additional given information - the solution is unique. It's extra and we don't need it to solve this equation. We're just using basic algebra to solve this problem.

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Красивое решение! Beautifull solution!

[prabhudasmandal]

Besides the conceptual solution, if we suppose (2x+3/2) ^2 as m and (2y--3) as n then we get (m^2+7/4) (n^2+16) =28.or m^2n^2+16m^2+7/4(n) ^2+28=28=0.additions of squares being=0, each part must be equal to zero. As such we get m=0, n=0.restoring the values, we get 2x+3/2=0 or x=--3/4 and when 2y--3=0, y=3/2.