Math Problem Solver

$$x^{2} - y^{2} + \frac{y^{2}}{x^{2}} = 9 - \frac{8 y^{2}}{9}$$

$$\frac{x}{y} = \frac{1}{3}$$Solve for $x$: $$x = \frac{y}{3}$$Substitute $x$ into the given expression: $$\begin{aligned}x^{2} - y^{2} + \frac{y^{2}}{x^{2}} &= - y^{2} + \frac{y^{2}}{\left(\frac{y}{3}\right)^{2}} + \left(\frac{y}{3}\right)^{2} \\ x^{2} - y^{2} + \frac{y^{2}}{x^{2}} &= 9 - \frac{8 y^{2}}{9}\end{aligned}$$