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The image shows a triangle where two sides are labeled as \(3x + 2\) and \(7x - 4\). The expression for the perimeter of the triangle is given as \(14x + 3\). We need to find the length of the third side of the triangle.

### Step-by-step solution:

1. The perimeter of a triangle is the sum of the lengths of all three sides.
2. Let the third side be denoted by \(S\).
3. According to the problem, the sum of all three sides equals the perimeter \(14x + 3\):
   \[
   (3x + 2) + (7x - 4) + S = 14x + 3
   \]
4. Combine the like terms from the known side lengths:
   \[
   3x + 7x + 2 - 4 + S = 14x + 3
   \]
   Simplifying:
   \[
   10x - 2 + S = 14x + 3
   \]
5. Solve for \(S\) by isolating it on one side:
   \[
   S = 14x + 3 - 10x + 2
   \]
   Simplifying:
   \[
   S = 4x + 5
   \]

### Final answer:
The length of the third side is \(4x + 5\).

Would you like more details or clarification on any part of this? Here are five related questions for further exploration:

1. How would you solve for \(x\) if given specific values for the perimeter?
2. Can you verify the result by plugging in values for \(x\)?
3. What would happen if the perimeter expression was different, say \(15x + 6\)?
4. How does the triangle's perimeter change if the length of the third side was altered?
5. Could you solve for the triangle's area if it was a right triangle?

**Tip:** When dealing with multiple variables in an equation, always combine like terms to simplify the problem.

The expression 14x + 3 represents the perimeter of the triangle. What is the length of the third side?

Learn how to find the length of the third side of a triangle when the perimeter is given as an algebraic expression. This problem involves solving linear equations and simplifying expressions. Suitable for students in Grades 7-9.

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