x+4x=y+4yx+4x=y+4y

⇔x2+4x=y2+4y⇔x2+4x=y2+4y

⇔y(x2+4)=x(y2+4)⇔y(x2+4)=x(y2+4)

⇒x2y+4y=xy2+4x⇒x2y+4y=xy2+4x

⇒x2y−xy2=4x−4y⇒x2y−xy2=4x−4y

⇒xy(x−y)=4(x−y)⇒xy(x−y)=4(x−y)

⇒xy(x−y)−4(x−y)=0⇒xy(x−y)−4(x−y)=0

⇒(xy−4)(x−y)=0⇒(xy−4)(x−y)=0

But x ≠ y 

⇒xy−4=0⇒xy−4=0

⇒xy=4

\(x+\dfrac{4}{x}=y+\dfrac{4}{y}\)

\(\Leftrightarrow\dfrac{x^2+4}{x}=\dfrac{y^2+4}{y}\)

\(\Leftrightarrow y\left(x^2+4\right)=x\left(y^2+4\right)\)

\(\Rightarrow x^2y+4y=xy^2+4x\)

\(\Rightarrow x^2y-xy^2=4x-4y\)

\(\Rightarrow xy\left(x-y\right)=4\left(x-y\right)\)

\(\Rightarrow xy\left(x-y\right)-4\left(x-y\right)=0\)

\(\Rightarrow\left(xy-4\right)\left(x-y\right)=0\)

But x ≠ y 

\(\Rightarrow xy-4=0\)

\(\Rightarrow xy=4\)