To solve the problem, let's analyze the information given in the image step-by-step:

1. Given Information:

   • $PR = 39$
   • We are asked to find $PQ$.

2. Diagram Analysis:

   • There is a line segment with three points: $P$, $Q$, and $R$.
   • The length from $P$ to $Q$ is represented by the expression $9x + 7$.
   • The length from $Q$ to $R$ is represented by the expression $-3x + 20$.
   • The total length $PR$ (from $P$ to $R$) is the sum of $PQ$ and $QR$.

3. Set Up the Equation: Since $PR = PQ + QR$, we can write: $9x + 7 + (-3x + 20) = 39$

4. Simplify the Equation: Combine like terms: $9x - 3x + 7 + 20 = 39$ $6x + 27 = 39$

5. Solve for $x$: Subtract 27 from both sides: $6x = 12$ Divide both sides by 6: $x = 2$

6. Find $PQ$: Substitute $x = 2$ back into the expression for $PQ$: $PQ = 9x + 7 = 9(2) + 7 = 18 + 7 = 25$

Conclusion:

The length of $PQ$ is 25.

Would you like more details or have any questions?

Further Related Questions:

1. How would the solution change if $PR$ had a different value?
2. Can you solve for $QR$ using the given information?
3. What would happen if the expressions for $PQ$ or $QR$ were nonlinear?
4. How do you solve similar problems involving unknowns in geometric figures?
5. What is the method for solving systems of equations derived from geometric diagrams?

Tip:

Always double-check your setup and ensure all expressions correctly represent the problem conditions before solving equations!