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Let's solve this step-by-step.

We are given the expression:

\[
2m + 6\left( \frac{2}{3} - m \right)
\]
where \( m = 4 \).

### Step 1: Simplify the expression

We need to distribute the 6 across the terms inside the parentheses:

\[
6\left( \frac{2}{3} - m \right) = 6 \times \frac{2}{3} - 6 \times m
\]

This becomes:

\[
6 \times \frac{2}{3} = 4 \quad \text{and} \quad 6 \times m = 6m
\]

So, the expression simplifies to:

\[
2m + 4 - 6m
\]

### Step 2: Combine like terms

Now, combine the \( m \)-terms:

\[
2m - 6m + 4 = -4m + 4
\]

Thus, the simplified expression is:

\[
-4m + 4
\]

### Step 3: Substitute \( m = 4 \) into the original and simplified expressions

- In the original expression:
  \[
  2m + 6\left( \frac{2}{3} - m \right)
  \]
  Substituting \( m = 4 \):
  \[
  2(4) + 6\left( \frac{2}{3} - 4 \right) = 8 + 6\left( \frac{2}{3} - 4 \right) = 8 + 6\left( \frac{2}{3} - \frac{12}{3} \right)
  \]
  \[
  = 8 + 6\left( -\frac{10}{3} \right) = 8 - 20 = -12
  \]

- In the simplified expression:
  \[
  -4m + 4
  \]
  Substituting \( m = 4 \):
  \[
  -4(4) + 4 = -16 + 4 = -12
  \]

### Final answers:
- The simplified expression is: \(-4m + 4\)
- The solution when substituting \( m = 4 \) is \(-12\).

Both the original and simplified expressions give the same result, so they are equivalent.

---

Would you like further details or explanations?

Here are 5 related questions to consider:
1. What happens if you substitute a different value for \( m \)?
2. How would the result change if the coefficient of \( m \) was different in the original expression?
3. Can you factor the simplified expression further?
4. How can you verify the equivalence of two algebraic expressions systematically?
5. What is the significance of distributing before simplifying in algebra?

**Tip:** Always combine like terms to simplify expressions effectively and reduce errors during substitution.

Simplify the expression 2m + 6(2/3 - m), where m = 4. Then substitute the given value for the variable in both expressions to determine if they are equivalent.

Learn how to simplify the algebraic expression 2m + 6(2/3 - m) and check if it is equivalent when substituting m = 4. This problem covers essential algebraic concepts, including the distributive property and substitution. Suitable for students in Grades 8-10.

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