What are the solutions to the equation \( x-\frac{7}{x}=6 ? \)

Solve the equation \( x-\frac{7}{x}=6 \). Solve the equation by following steps: - step0: Solve for \(x\): \(x-\frac{7}{x}=6\) - step1: Find the domain: \(x-\frac{7}{x}=6,x\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\left(x-\frac{7}{x}\right)x=6x\) - step3: Simplify the equation: \(x^{2}-7=6x\) - step4: Move the expression to the left side: \(x^{2}-7-6x=0\) - step5: Factor the expression: \(\left(x-7\right)\left(x+1\right)=0\) - step6: Separate into possible cases: \(\begin{align}&x-7=0\\&x+1=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=7\\&x=-1\end{align}\) - step8: Check if the solution is in the defined range: \(\begin{align}&x=7\\&x=-1\end{align},x\neq 0\) - step9: Find the intersection: \(\begin{align}&x=7\\&x=-1\end{align}\) - step10: Rewrite: \(x_{1}=-1,x_{2}=7\) The solutions to the equation \(x-\frac{7}{x}=6\) are \(x=-1\) and \(x=7\).