The three sides of a triangular fence have lengths 3x + 4, y − 5, and 7x − 1. What is the total perimeter?

10x + y − 2 10x + y + 8 10x + y 3x + y − 2

The perimeter of a triangle is found by adding the lengths of all three sides. In this case, the three sides have the following expressions:

1. \( 3x + 4 \)
2. \( y - 5 \)
3. \( 7x - 1 \)

The total perimeter \( P \) will be the sum of these three expressions:

\[
P = (3x + 4) + (y - 5) + (7x - 1)
\]

Now, let's combine like terms:

- Combine the terms with \( x \): \( 3x + 7x = 10x \)
- Combine the constant terms: \( 4 - 5 - 1 = -2 \)
- Combine the terms with \( y \): \( y \)

So, the total perimeter is:

\[
P = 10x + y - 2
\]

Thus, the correct answer is: **\( 10x + y - 2 \)**.

Would you like any further clarification or additional steps on this? 

Here are some related questions for you:
1. How would the perimeter change if one of the sides were a constant value (e.g., 10)?
2. What is the relationship between the perimeter and the individual side lengths of a triangle?
3. If we know the value of \( x \) and \( y \), how would we calculate the exact perimeter?
4. How do the perimeter calculations differ for other polygons (e.g., quadrilaterals or pentagons)?
5. How do you solve for the sides of a triangle if you're given the perimeter and expressions like the ones above?

**Tip:** When calculating the perimeter or sum of several expressions, always ensure you combine like terms (e.g., terms involving \( x \) with other \( x \)-terms, constants with constants).

The three sides of a triangular fence have lengths 3x + 4, y − 5, and 7x − 1. What is the total perimeter?

Learn how to calculate the perimeter of a triangle when given algebraic expressions for its sides. This problem involves simplifying expressions to find the total perimeter, focusing on combining like terms in algebra. Suitable for students in Grades 7-9.

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