Question
Solve the system
x+y+\dfrac{1}{x}+\dfrac{1}{y} = \dfrac{9}{2} \\ xy+\dfrac{1}{xy} =\dfrac{5}{2}
Solve the system
x+y+\dfrac{1}{x}+\dfrac{1}{y} = \dfrac{9}{2} \\ xy+\dfrac{1}{xy} =\dfrac{5}{2}
Let
a=xy
b=x+y
The two equations are transformed to
Solve (2) for a
a^2-\dfrac{5}{2}a+1=0
Then we get
a=\dfrac{1}{2} or a=2
Substituting the value of a to (1) gives
b=\dfrac{3}{2} or b = 3
Then we have two cases for x,y
and
Case 1, x,y is the roots for the quadratic equation z^2-\dfrac{3}{2}z+\dfrac{1}{2} =0 based on Vieta's formula.
Solving the equation gives two solutions for x,y
\begin{cases} x=\dfrac{1}{2} \\ y=1 \end{cases} \quad\text{or}\quad \begin{cases} x=1 \\ y=\dfrac{1}{2} \end{cases}
Case 2, x,y is the roots for the quadratic equation z^2-3z+2 =0 based on Vieta's formula. Then we get another 2 solutions.
\begin{cases} x=1\\ y=2 \end{cases} \quad\text{or}\quad \begin{cases} x=2 \\ y=1 \end{cases}
In summary, there're four solutions for the system of equations.