Solve the rational equation: \( \frac{x-8}{x}=\frac{63}{x+8} \)

Solve the equation \( \frac{x-8}{x}=\frac{63}{x+8} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x-8}{x}=\frac{63}{x+8}\) - step1: Find the domain: \(\frac{x-8}{x}=\frac{63}{x+8},x \in \left(-\infty,-8\right)\cup \left(-8,0\right)\cup \left(0,+\infty\right)\) - step2: Cross multiply: \(\left(x-8\right)\left(x+8\right)=x\times 63\) - step3: Simplify the equation: \(\left(x-8\right)\left(x+8\right)=63x\) - step4: Expand the expression: \(x^{2}-64=63x\) - step5: Move the expression to the left side: \(x^{2}-64-63x=0\) - step6: Factor the expression: \(\left(x-64\right)\left(x+1\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x-64=0\\&x+1=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=64\\&x=-1\end{align}\) - step9: Check if the solution is in the defined range: \(\begin{align}&x=64\\&x=-1\end{align},x \in \left(-\infty,-8\right)\cup \left(-8,0\right)\cup \left(0,+\infty\right)\) - step10: Find the intersection: \(\begin{align}&x=64\\&x=-1\end{align}\) - step11: Rewrite: \(x_{1}=-1,x_{2}=64\) The solutions to the rational equation \( \frac{x-8}{x}=\frac{63}{x+8} \) are \( x=-1 \) and \( x=64 \).