Question

which expression is equivalent to the expression \\(\frac{2}{3}(9z + 6) - 7z\\)? \\(\circ\\ 4 - z\\) \\(\circ\\ \frac{4z}{3} + 4\\) \\(\circ\\ 6z + 6 - \frac{14z}{3}\\) \\(\circ\\ \frac{18z}{3} - \frac{12}{3} - \frac{14z}{3}\\)

Explanation:

Step1: Distribute the fraction

First, we distribute \(\frac{2}{3}\) to the terms inside the parentheses \(9z + 6\). Using the distributive property \(a(b + c)=ab+ac\), we have:
\(\frac{2}{3}(9z + 6)=\frac{2}{3}\times9z+\frac{2}{3}\times6\)
Calculating each term: \(\frac{2}{3}\times9z = 6z\) and \(\frac{2}{3}\times6 = 4\). So the expression becomes \(6z + 4-7z\).

Step2: Combine like terms

Now, we combine the like terms \(6z\) and \(-7z\). Combining like terms means adding their coefficients. The coefficient of \(6z\) is \(6\) and the coefficient of \(-7z\) is \(-7\). So \(6z-7z=(6 - 7)z=-z\). And we still have the constant term \(4\). So the simplified expression is \(4 - z\).

Answer:

\(4 - z\) (the first option among the given choices)