Question

read and try to solve the problem below.
are $\frac{1}{3}(-3m + 6 - 12 + 15m)$ and $2(1 - 2m)$ equivalent expressions?
show why or why not.

Explanation:

Step1: Simplify the first expression

First, simplify \(\frac{1}{3}(-3m + 6 - 12 + 15m)\). Combine like terms inside the parentheses: \(-3m + 15m = 12m\) and \(6 - 12 = -6\). So the expression inside the parentheses becomes \(12m - 6\). Now distribute \(\frac{1}{3}\): \(\frac{1}{3}(12m - 6) = \frac{1}{3} \times 12m - \frac{1}{3} \times 6 = 4m - 2\).

Step2: Simplify the second expression

Simplify \(2(1 - 2m)\) by distributing the 2: \(2 \times 1 - 2 \times 2m = 2 - 4m\).

Step3: Compare the two simplified expressions

The first expression simplifies to \(4m - 2\) and the second to \(2 - 4m\). These are not the same because the coefficients of \(m\) are different (4 vs. -4) and the constant terms are also different (-2 vs. 2) when considering the form. For example, if we set \(m = 0\), the first expression is \(-2\) and the second is \(2\), which are not equal.

Answer:

\(\frac{1}{3}(-3m + 6 - 12 + 15m)\) simplifies to \(4m - 2\) and \(2(1 - 2m)\) simplifies to \(2 - 4m\). Since \(4m - 2
eq 2 - 4m\) (e.g., when \(m = 0\), \(4(0) - 2 = -2\) and \(2 - 4(0) = 2\), and \(-2
eq 2\)), the two expressions are not equivalent.