#### Question

Solve the system

x+y+\dfrac{1}{x}+\dfrac{1}{y} = \dfrac{9}{2} \\ xy+\dfrac{1}{xy} =\dfrac{5}{2}

Collected in the board: System of equations

Steven Zheng posted 3 minutes ago

Let

a=xy

b=x+y

The two equations are transformed to

b+\dfrac{b}{a} = \dfrac{9}{2}
(1)
a+\dfrac{1}{a} =\dfrac{5}{2}
(2)

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

\begin{cases} xy=\dfrac{1}{2} \\ x+y=\dfrac{3}{2} \end{cases}
(3)

and

\begin{cases} xy=2 \\ x+y = 3 \end{cases}
(4)

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