https://file.aiquickdraw.com/mathbot/file/1726793207882cauvv6t6.jpg

Let's solve the given equation step by step.

The equation is:

\[
-9v + 31 = 2(v - 1)
\]

### Step 1: Distribute on the right-hand side

First, distribute the 2 on the right-hand side:

\[
-9v + 31 = 2v - 2
\]

### Step 2: Move all terms with \(v\) to one side and constants to the other

To isolate \(v\), subtract \(2v\) from both sides:

\[
-9v - 2v + 31 = -2
\]

This simplifies to:

\[
-11v + 31 = -2
\]

Next, subtract 31 from both sides:

\[
-11v = -2 - 31
\]

\[
-11v = -33
\]

### Step 3: Solve for \(v\)

Now, divide both sides by \(-11\):

\[
v = \frac{-33}{-11} = 3
\]

So, the solution is:

\[
v = 3
\]

Would you like more details on any of these steps, or do you have any questions?

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Here are five related questions to consider:
1. How would the steps change if the equation was \( -9v + 31 = 2(v + 1) \)?
2. What if instead of \( -9v \), the equation started with \( -5v \)? How would that affect the solution?
3. Can you rewrite the equation as \( 9v = 2(1 - v) - 31 \) and solve for \(v\)?
4. How do you check your solution once you’ve solved for \(v\)?
5. What happens to the solution if all the signs in the equation were reversed?

**Tip:** When solving linear equations, always aim to simplify the equation step by step to avoid errors!

Learn how to solve the linear equation -9v + 31 = 2(v - 1) with a detailed step-by-step solution. This problem covers key algebraic concepts, including distribution and isolating variables. Suitable for students in Grades 7-9.

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