Solve the equation. What is the value of v?

Question

One way you can start solving is to make sure there are only v terms on one side. What move can you make to both sides of the equation to get rid of the
4
3
​
v on the left, while keeping the equation balanced?
4
3
​
v + 8
=
v + 12

Multiply both sides by
4
3
​
v

Add
4
3
​
v to both
sides

Subtract
4
3
​
v from both
sides

Divide both sides by
4
3
​
v
Nice!
You can subtract
4
3
​
v from the left side of the equation, so that the v terms are only on the right side of the equation, while keeping the equation balanced. Record subtracting
4
3
​
v from both sides now.
4
3
​
v + 8
=
v + 12
−

4
3
​
v
−

4
3
​
v
Great job!
We have v terms on both sides of the equation, and we want to get the v terms on one side. We can subtract
4
3
​
v on the left side of the equation so that there are no v terms on the left. Then we need to subtract
4
3
​
v from the right to keep the equation balanced. Go for it.
4
3
​
v + 8
=
v + 12
−
4
3
​
v
−
4
3
​
v

=
Great job!
Keep going. Let's try to get
4
1
​
v alone on the right side. How could you get rid of the 12 on the right side so that
4
1
​
v is alone on the right, while keeping the equation balanced?
4
3
​
v + 8
=
v + 12
−
4
3
​
v
−
4
3
​
v

8
=
4
1
​
v + 12

Divide both sides by 12

Add 12 to both sides

Subtract 12 from both sides

Answer 1

Let's start from where we were after the steps you outlined. The equation we had was:

\[ 8 = v + 12 - \frac{4}{3}v \]

Now, we need to simplify the right side by combining the \(v\) terms. By rearranging and simplifying the equation, we can write it as:

\[ 8 = v - \frac{4}{3}v + 12 \]

Now, we express \(v\) terms as:

\[ v - \frac{4}{3}v = \frac{3}{3}v - \frac{4}{3}v = -\frac{1}{3}v \]

So the equation simplifies to:

\[ 8 = -\frac{1}{3}v + 12 \]

Next, we want to eliminate the \(12\) on the right side. We can do this by subtracting \(12\) from both sides:

\[ 8 - 12 = -\frac{1}{3}v \]

This simplifies to:

\[ -4 = -\frac{1}{3}v \]

Now, to get \(v\) alone, we can multiply both sides by \(-3\):

\[ -4 \times -3 = v \]

Thus:

\[ v = 12 \]

So the final answer is:

\[ \boxed{12} \]