What is the simplified expression for the given expression?
$\frac{1}{2}(8 x+4)+rac{1}{3}(9-3 x)$
To simplify the expression \(\frac{1}{2}(8x+4) + \frac{1}{3}(9-3x)\), we apply the distributive property, which states that \(a(b+c) = ab + ac\). We process each part of the sum separately. The goal is to reach a simplified linear expression in the form \(Ax + B\).
First, we distribute \(\frac{1}{2}\) into the binomial \((8x+4)\):
\[\frac{1}{2}(8x+4) = \left(\frac{1}{2} \times 8x\right) + \left(\frac{1}{2} \times 4\right) = 4x + 2.\]
Next, we distribute \(\frac{1}{3}\) into the binomial \((9-3x)\):
\[\frac{1}{3}(9-3x) = \left(\frac{1}{3} \times 9\right) - \left(\frac{1}{3} \times 3x\right) = 3 - x.\]
Now, we combine the simplified parts to get the final expression:
\[(4x + 2) + (3 - x).\]
Group the terms with \(x\) and the constant terms together:
\[(4x - x) + (2 + 3) = 3x + 5.\]
Comparing this result with the given choices, we see that it matches option a.
Simplifying algebraic expressions is fundamental in many real-world applications, such as budget management or engineering. For example, if you are buying supplies for a project and need to calculate costs for different quantities of items described by linear models, you can simplify these models to quickly find the total cost. If one component's cost is modeled by \(\frac{1}{2}(8x+4)\) and another by \(\frac{1}{3}(9-3x)\), simplifying their sum helps you immediately determine the total cost \(3x+5\) based on the variable \(x\), allowing for efficient financial planning.