A wire of length (5𝑎𝑏 − 2𝑏 + 1) m is cut into three small parts. If the length of the two small parts is (3𝑎𝑏 + 𝑏) m in total, find the length of another piece of the wire.

To solve the problem, we can start by defining the length of the wire and the length of the parts.

Given:
- Total length of the wire = \( 5ab - 2b + 1 \) meters.
- Length of two small parts combined = \( 3ab + b \) meters.

Let the length of the third part be \( x \) meters.

According to the problem, the sum of the lengths of the three parts must equal the total length of the wire. Therefore, we can set up the following equation:

\[
(3ab + b) + x = 5ab - 2b + 1
\]

To find the length of the third part, solve for \( x \):

\[
x = (5ab - 2b + 1) - (3ab + b)
\]

Now, simplify the equation by distributing and combining like terms:

\[
x = 5ab - 2b + 1 - 3ab - b
\]

\[
x = (5ab - 3ab) + (-2b - b) + 1
\]

\[
x = 2ab - 3b + 1
\]

So, the length of the third piece of wire is \( 2ab - 3b + 1 \) meters.

Would you like more details, or do you have any questions?

Here are 8 related questions for further exploration:
1. What is the length of the third piece if \( a = 2 \) and \( b = 1 \)?
2. How would the solution change if the wire was cut into four parts instead of three?
3. If \( b \) were negative, how would that affect the length of the third part?
4. Can you verify the solution by substituting specific values for \( a \) and \( b \)?
5. What would happen if one of the parts was doubled in length?
6. How would the length change if the total length of the wire was \( 6ab - 3b + 2 \) meters?
7. What is the relationship between the lengths of the parts and the total length of the wire?
8. How can we generalize this problem for different numbers of wire segments?

**Tip:** When dealing with algebraic expressions, always combine like terms to simplify the problem-solving process.

Discover how to find the length of the third piece of wire when a wire of length (5ab - 2b + 1) meters is cut into three parts. Includes step-by-step solution and detailed algebraic approach. Suitable for Grades 9-12 students studying algebra and equations.

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