Example 1 If the equations of the two asymptotes of a hyperbola are $y= \pm \frac{2}{3} x$, and it passes through the point $M\left(\frac{9}{2},-1\right)$, try to find its equation.
Solution: The equation of the hyperbola system with asymptotes $y= \pm \frac{2}{3} x$, i.e., $\frac{x^{2}}{3^{2}}-\frac{y^{2}}{2^{2}}=0$, is $\frac{x^{2}}{3^{2}}-\frac{y^{2}}{2^{2}}=\lambda(\lambda \neq 0)$.
Given that point $M\left(\frac{9}{2},-1\right)$ lies on it, substituting gives $\lambda=2$.
Therefore, the equation of the required hyperbola is $\frac{x^{2}}{18}-\frac{y^{2}}{8}=1$.
Comment: The equation of the hyperbola system with the same asymptotes as $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=\lambda(\lambda \neq 0)$.