Question

solve for n.
\\(\frac{n + 1}{n + 10} = \frac{n - 1}{n + 6}\\)

Explanation:

Step1: Cross - multiply the fractions

To solve the equation \(\frac{n + 1}{n+10}=\frac{n - 1}{n + 6}\), we use the cross - multiplication property of fractions. If \(\frac{a}{b}=\frac{c}{d}\) (where \(b
eq0\) and \(d
eq0\)), then \(a\times d=c\times b\). So we get \((n + 1)(n + 6)=(n - 1)(n + 10)\).

Step2: Expand both sides

Expand the left - hand side: \((n + 1)(n + 6)=n\times n+n\times6 + 1\times n+1\times6=n^{2}+6n+n + 6=n^{2}+7n + 6\).
Expand the right - hand side: \((n - 1)(n + 10)=n\times n+n\times10-1\times n-1\times10=n^{2}+10n - n-10=n^{2}+9n - 10\).

Step3: Subtract \(n^{2}\) from both sides

Subtract \(n^{2}\) from both sides of the equation \(n^{2}+7n + 6=n^{2}+9n - 10\). We have \(n^{2}-n^{2}+7n + 6=n^{2}-n^{2}+9n - 10\), which simplifies to \(7n+6 = 9n-10\).

Step4: Isolate the variable \(n\)

Subtract \(7n\) from both sides: \(7n - 7n+6=9n - 7n-10\), so \(6 = 2n-10\).
Then add 10 to both sides: \(6 + 10=2n-10 + 10\), which gives \(16 = 2n\).

Step5: Solve for \(n\)

Divide both sides by 2: \(\frac{16}{2}=\frac{2n}{2}\), so \(n = 8\).

Answer:

\(n = 8\)