A 45-foot wire is cut into three pieces. The second piece is 3 times as long as the first piece. The third piece is 12 feet longer than the first piece.

Let's denote the length of the first piece of wire as \( x \) feet.

According to the problem:

- The length of the second piece is \( 3x \) (3 times the first piece).
- The length of the third piece is \( x + 12 \) (12 feet longer than the first piece).

Since the total length of the wire is 45 feet, you can set up the following equation:

\[
x + 3x + (x + 12) = 45
\]

Now, simplify the equation:

\[
x + 3x + x + 12 = 45
\]

Combine like terms:

\[
5x + 12 = 45
\]

This equation can be used to find the length of the first piece.

There are more equations than that, had to clarify. What equation(s) can be used to find the length of the first piece?

To find the length of the first piece of the wire, we can derive multiple equations from the problem statements:

1. **Define Variables:**
- Let \( x \) be the length of the first piece (in feet).
- The second piece is \( 3x \) (3 times as long as the first piece).
- The third piece is \( x + 12 \) (12 feet longer than the first piece).

2. **Write the Equation for Total Length:**
The total length of the wire adds up to 45 feet, which gives us the equation:
\[
x + 3x + (x + 12) = 45
\]

3. **Simplify the Total Length Equation:**
Combine like terms:
\[
5x + 12 = 45
\]

4. **Rearrangement for Simplicity:**
To find the length of the first piece, we can rearrange the equation:
\[
5x = 45 - 12
\]
\[
5x = 33
\]
\[
x = \frac{33}{5} = 6.6 \text{ feet}
\]

### Additional Conceptual Formulations:
You could also express this scenario in a different way:

1. **Equations for Each Piece:**
- First piece: \( L_1 = x \)
- Second piece: \( L_2 = 3x \)
- Third piece: \( L_3 = x + 12 \)

2. **Equation for the Lengths of Each Piece:**
The individual pieces can also lead to the same equation:
\[
L_1 + L_2 + L_3 = 45
\]
or explicitly:
\[
x + 3x + (x + 12) = 45
\]

### Summary:
In summary, the main equation to solve for the first piece's length is:
\[
5x + 12 = 45
\]
This suffices to derive that \( x = 6.6 \) feet, and this methodology allows you to understand the relationships between the pieces clearly.

Select all that apply

p+3+p+p+12=45

3p+15=45

p+3p+p+12=45

5p+12=45

To determine which equations are correct representations based on the problem, let's analyze the equations provided:

1. **Equation: \( p + 3 + p + p + 12 = 45 \)**
- This equation is **not correct** because it does not properly account for the lengths of the three pieces. It does not maintain the relationship that the second piece is \( 3 \) times the first and the third is \( 12 \) feet longer than the first.

2. **Equation: \( 3p + 15 = 45 \)**
- This equation is **not correct** as it seems to be manipulating terms without isolating the variables in the problem as stated.

3. **Equation: \( p + 3p + p + 12 = 45 \)**
- This equation is **correct** as it accurately represents the relationship \( (x) + (3x) + (x + 12) = 45 \). Simplifying this will lead to a proper representation of the problem.

4. **Equation: \( 5p + 12 = 45 \)**
- This is also **correct** as it is the result of simplifying \( p + 3p + (p + 12) = 45 \) to form \( 5p + 12 = 45 \).

### Summary:
The correct equations that apply based on the problem are:

- **\( p + 3p + p + 12 = 45 \)**
- **\( 5p + 12 = 45 \)**

The other two equations do not correctly represent the situation described.