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.