Question

1. \\(\frac{5}{n} = \frac{6}{n^2} + \frac{n + 5}{n^2}\\)

Explanation:

Step1: Combine right - hand side fractions

Since the two fractions on the right - hand side have the same denominator \(n^{2}\), we can add them according to the rule \(\frac{a}{c}+\frac{b}{c}=\frac{a + b}{c}\). So \(\frac{6}{n^{2}}+\frac{n + 5}{n^{2}}=\frac{6+(n + 5)}{n^{2}}=\frac{n+11}{n^{2}}\). Now our equation becomes \(\frac{5}{n}=\frac{n + 11}{n^{2}}\).

Step2: Eliminate the denominators

We can multiply both sides of the equation by \(n^{2}\) (assuming \(n
eq0\), because if \(n = 0\), the original equation has undefined terms) to get rid of the denominators. Multiplying the left - hand side \(\frac{5}{n}\times n^{2}=5n\), and multiplying the right - hand side \(\frac{n + 11}{n^{2}}\times n^{2}=n + 11\). So the equation is now \(5n=n + 11\).

Step3: Solve for \(n\)

Subtract \(n\) from both sides of the equation \(5n-n=n + 11 - n\). This simplifies to \(4n=11\). Then divide both sides by 4: \(n=\frac{11}{4}\).

Answer:

\(n=\frac{11}{4}\)