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Let's solve the equation \(-2a + 3b = -6\) for \(b\):

### Step 1: Isolate the term with \(b\)
We want to solve for \(b\), so first, move the term with \(a\) to the other side by adding \(2a\) to both sides:

\[
3b = 2a - 6
\]

### Step 2: Solve for \(b\)
Now, divide both sides by 3 to isolate \(b\):

\[
b = \frac{2a - 6}{3}
\]

### Step 3: Simplify the equation
We can simplify the expression:

\[
b = \frac{2a}{3} - \frac{6}{3}
\]

\[
b = \frac{2a}{3} - 2
\]

So, the correct solution is:

\[
b = \frac{2a}{3} - 2
\]

This matches the first option in the list: 
\[
b = \frac{2}{3}a - 2
\]

Thus, the correct answer is: **\( b = \frac{2a}{3} - 2 \)**.

Would you like more details on any of the steps?

Here are 5 related questions for further practice:
1. How would you solve for \(a\) if the equation were \(3a - 4b = 8\)?
2. How would the solution change if the equation were \(5a + 2b = 10\)?
3. What happens if we have fractions in the original equation?
4. How can you check your solution for a linear equation like this one?
5. What are the key steps in isolating variables in any equation?

**Tip:** Always double-check your operations, especially when dealing with negatives and fractions.

Discover how to solve the linear equation -2a + 3b = -6 for b, using algebraic manipulation to isolate the variable. This problem is a great practice for understanding variable isolation in linear equations, suitable for middle to high school students.

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