Question

1. $1 - \frac{6n + 36}{n} = \frac{1}{3}$

Explanation:

Step1: Eliminate denominator

Multiply both sides by \(3n\) (LCM of \(n\) and \(3\)):
\(3n\times1 - 3n\times\frac{6n + 36}{n}=3n\times\frac{1}{3}\)
Simplify: \(3n - 3(6n + 36)=n\)

Step2: Expand and simplify

Expand the left side: \(3n - 18n - 108 = n\)
Combine like terms: \(-15n - 108 = n\)

Step3: Solve for \(n\)

Subtract \(n\) from both sides: \(-16n - 108 = 0\)
Add \(108\) to both sides: \(-16n = 108\)
Divide by \(-16\): \(n = -\frac{108}{16}=-\frac{27}{4}\)

Step4: Check validity

Denominator \(n = -\frac{27}{4}
eq0\), so it is valid.

Answer:

\(n = -\dfrac{27}{4}\)