To solve for $PR$, we can use the fact that the sum of the segments $PQ$ and $QR$ equals the total segment $PR$. Therefore, we have:

$PQ + QR = PR$

Substituting the given expressions:

$(5x - 13) + (6x + 7) = 9x + 12$

Let's combine the terms on the left side:

$5x - 13 + 6x + 7 = 9x + 12$

Simplify:

$11x - 6 = 9x + 12$

Next, we'll move all terms involving $x$ to one side and constant terms to the other side:

$11x - 9x = 12 + 6$

$2x = 18$

Solve for $x$:

$x = 9$

Now that we have $x = 9$, we can substitute it back into the expression for $PR$:

$PR = 9x + 12 = 9(9) + 12 = 81 + 12 = 93$

So, $PR = 93$.

Would you like any further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would the solution change if the equation for $QR$ were different?
2. What is the importance of verifying the equations in geometry problems?
3. How do you approach solving equations that involve multiple variables?
4. Can the same method be applied to non-linear equations?
5. How do you interpret geometric segments algebraically?

Tip: When solving algebraic equations, always remember to check your solution by substituting it back into the original equation to ensure it satisfies all conditions.