Question

Solve the equation. Check your solution.
4 left parenthesis 4 plus v right parenthesis plus 34 v equals 15 v4(4+v)+34v=15v

Answer 1

To solve the equation \( 4(4 + v) + 34v = 15v \), we will first simplify and rearrange the equation.

1. Distribute \(4\) on the left side:
\[
4 \cdot 4 + 4v + 34v = 15v
\]
This expands to:
\[
16 + 4v + 34v = 15v
\]

2. Combine the like terms:
\[
16 + 38v = 15v
\]

3. Now, let's isolate \(v\). First, subtract \(15v\) from both sides:
\[
16 + 38v - 15v = 0
\]
This simplifies to:
\[
16 + 23v = 0
\]

4. Now, subtract \(16\) from both sides:
\[
23v = -16
\]

5. Finally, divide by \(23\):
\[
v = -\frac{16}{23}
\]

Now we have the solution \( v = -\frac{16}{23} \).

**To check the solution**, substitute \(v\) back into the original equation:

1. Substitute \(v = -\frac{16}{23}\) into \(4(4 + v) + 34v\):
\[
4\left(4 - \frac{16}{23}\right) + 34\left(-\frac{16}{23}\right)
\]

First calculate \(4 + v\):
\[
4 - \frac{16}{23} = \frac{92}{23} - \frac{16}{23} = \frac{76}{23}
\]

Now, calculate \(4(4 + v)\):
\[
4 \cdot \frac{76}{23} = \frac{304}{23}
\]

And calculate \(34v\):
\[
34 \cdot -\frac{16}{23} = -\frac{544}{23}
\]

Add these two results:
\[
\frac{304}{23} - \frac{544}{23} = \frac{304 - 544}{23} = \frac{-240}{23}
\]

2. Next, check \(15v\):
\[
15v = 15 \cdot -\frac{16}{23} = -\frac{240}{23}
\]

Now, we compare:
\[
\frac{-240}{23} \text{ (LHS) } = -\frac{240}{23} \text{ (RHS) }
\]

Since both sides are equal, our solution is confirmed correct.

**Final answer:**
\[
v = -\frac{16}{23}
\]