Question

If \(5+n=9-\frac{1}{3} n\), what is the value of \(n\) ? A) 3 B) 4 C) 6 D) \(\frac{21}{2}\)

Short answer

A) 3

Step-by-step solution

Add \(\frac{1}{3}n\) to both sides of the equation

To isolate the variable \(n\), we need to add \(\frac{1}{3}n\) to both sides of the equation. This will eliminate the term \(-\frac{1}{3}n\) from the right-hand side of the equation. \[5 + n + \frac{1}{3}n = 9 - \frac{1}{3}n + \frac{1}{3}n\]

Simplify the equation

Now we need to simplify the equation by combining the terms with the variable \(n\). \[\frac{1}{3}n + n = \frac{1}{3}n + \frac{3}{3}n = \frac{4}{3}n\] So, the simplified equation becomes: \[5 + \frac{4}{3}n = 9\]

Subtract 5 from both sides of the equation

Next, we need to subtract 5 from both sides of the equation to further isolate the variable \(n\). \[\frac{4}{3}n = 9 - 5\]

Simplify the equation again

Now we can simplify the right side of the equation: \[\frac{4}{3}n = 4\]

Divide by \(\frac{4}{3}\) to solve for \(n\)

Finally, we need to divide both sides of the equation by \(\frac{4}{3}\) to solve for the value of \(n\). \[n = \frac{4}{\frac{4}{3}}\]

Simplify the fraction to find the value of \(n\)

To simplify the fraction, multiply the numerator and the denominator by the reciprocal of the denominator: \[n = \frac{4}{1} \times \frac{3}{4}\] This simplifies to: \[n = 3\] So, the value of \(n\) is 3, which corresponds to choice A.