https://file.aiquickdraw.com/mathbot/file/1726710618778uvyzprqm.jpg

I can see the image contains several mathematical problems, mostly involving equations related to line segments PQ, QR, and PR. Let me help you solve these one by one.

1. **Problem 1**: 
   \[
   PQ = 3x + 14 \quad \text{and} \quad QR = 7x - 10
   \]
   To find \( x \), typically, you would be given a relation such as \( PQ = QR \). If that's the case, we can solve:
   \[
   3x + 14 = 7x - 10
   \]
   Let's solve for \( x \).

2. **Problem 2**: 
   \[
   PQ = 2x + 1 \quad \text{and} \quad QR = 5x - 44
   \]
   We are asked to find \( PQ \). You may need to solve for \( x \) first using the relation \( PQ = QR \), then substitute \( x \) back into \( PQ \).

3. **Problem 3**:
   \[
   PQ = 6x + 25 \quad \text{and} \quad QR = 16 - 3x
   \]
   Here, you might need to find \( PR \) by solving a similar equation between \( PQ \) and \( QR \).

4. **Problem 4**: 
   \[
   PR = 9x - 31 \quad \text{and} \quad QR = 43
   \]
   To find \( x \), we may have \( PR = QR + \text{(another relation)} \).

I'll now solve the first problem to show the steps.For the first problem, solving the equation \( 3x + 14 = 7x - 10 \) gives \( x = 6 \).

Would you like me to proceed with solving the other problems? Let me know if you need further explanation on this one!

Here are 5 related questions to explore:

1. How do you verify the solution for \( x \) once you find it?
2. What happens if you substitute \( x = 6 \) into the original \( PQ \) and \( QR \) expressions?
3. Could there be cases where \( PQ \neq QR \)? How would that change the approach?
4. How does this equation relate to line segments geometrically?
5. What strategies help simplify solving equations with variables on both sides?

**Tip**: Always check your solution by substituting it back into the original equation to ensure accuracy.

Solve for x in the following algebraic equations: PQ = 3x + 14 and QR = 7x - 10; PQ = 2x + 1 and QR = 5x - 44; PQ = 6x + 25 and QR = 16 - 3x; PR = 9x - 31 and QR = 43.

Solve a set of algebraic equations involving the relationships between line segments PQ, QR, and PR. This includes finding x by solving linear expressions for different scenarios. Suitable for students in Grades 8-10.

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