Question

Complete the following stepped out solutions to show that negative 3 ( - 1/3 + m) + 11 is equivalent to -3 • m + 12. -3 (-1/3 + m) + 11 - 3 • blank + ( - 3 • m) + 11 apply the distributive property blank + ( - 3 • m) + 11 apply the inverse property of multiplication. - 3 • m + blank + 11 apply the commutative property of addition and add - 3 • m + 12

Answer 1

Let's complete the solution step by step to show that \(-3\left(-\frac{1}{3} + m\right) + 11\) is equivalent to \(-3 \cdot m + 12\).

1. Start with the original expression:
\[
-3\left(-\frac{1}{3} + m\right) + 11
\]

2. Apply the distributive property:
\[
-3 \cdot \left(-\frac{1}{3}\right) + (-3 \cdot m) + 11
\]
- This means we distribute \(-3\) to both terms inside the parentheses. In this case:
\(-3 \cdot -\frac{1}{3} = 1\) and \(-3 \cdot m = -3m\).

3. Now we can rewrite the expression:
\[
1 + (-3 \cdot m) + 11
\]

4. Apply the inverse property of multiplication (which we essentially already did in step 2 with \(-3 \cdot m\)):
\[
1 - 3m + 11
\]

5. Next, combine like terms (i.e., the constant terms):
\[
1 + 11 - 3m = 12 - 3m
\]

6. Finally, rearrange using the commutative property of addition:
\[
-3m + 12
\]

So the complete expression shows:
\[
-3\left(-\frac{1}{3} + m\right) + 11 \equiv -3 \cdot m + 12
\]

Here are the blanks filled in:

- First blank: \(-3 \cdot -\frac{1}{3}\)
- Second blank: \(1\)
- Third blank: \(12\)

Final result:
- The overall equivalence after steps:
\[
-3(-\frac{1}{3} + m) + 11 \rightarrow 1 + (-3 \cdot m) + 11 \rightarrow -3 \cdot m + 12
\]