What is the value of $n$ in the equation $\frac{1}{2}(n-4)-3=3-(2 n+3)$?
The given equation is \(\frac{1}{2}(n-4) - 3 = 3 - (2n + 3)\). Our objective is to isolate the variable \(n\) to find its value. We will perform this by simplifying both sides of the equation using the distributive property, combining like terms, and then isolating \(n\). This is a linear equation in one variable, \(n\).
First, expand the expression on both sides:
On the left side:
\(\frac{1}{2}(n-4) - 3 = \frac{1}{2}n - \frac{1}{2}(4) - 3 = \frac{1}{2}n - 2 - 3 = \frac{1}{2}n - 5\)
On the right side:
\(3 - (2n + 3) = 3 - 2n - 3 = -2n\)
The equation simplifies to:
\(\frac{1}{2}n - 5 = -2n\)
Now, move all terms containing \(n\) to the left side and the constants to the right side by adding \(2n\) to both sides and adding \(5\) to both sides:
\(\frac{1}{2}n + 2n = 5\)
To add the terms with \(n\), express \(2n\) as \(\frac{4}{2}n\):
\(\frac{1}{2}n + \frac{4}{2}n = 5\)
\(\frac{5}{2}n = 5\)
To solve for \(n\), multiply both sides by \(\frac{2}{5}\):
\(n = 5 \times \frac{2}{5} = 2\)
We found \(n = 2\). To verify, substitute \(n=2\) back into the original equation:
Left Side: \(\frac{1}{2}(2 - 4) - 3 = \frac{1}{2}(-2) - 3 = -1 - 3 = -4\)
Right Side: \(3 - (2(2) + 3) = 3 - (4 + 3) = 3 - 7 = -4\)
Since both sides are equal, \(n=2\) is the correct solution, which corresponds to the second option. The final answer is \(\boxed{2}\).
In real life, solving linear equations like this one is useful in many situations. For instance, if you are comparing two different payment plans for a service, say Plan A charges a fixed fee plus a cost per unit, and Plan B charges a different fixed fee plus a cost per unit, you can write an equation similar to this one to find the number of units \(n\) for which both plans cost exactly the same. Mastering this simple algebraic technique allows you to make informed decisions about efficiency and cost-effectiveness.