https://file.aiquickdraw.com/mathbot/file/17267784784467c0jsrhy.jpg

The problem states that T is the midpoint of line segment \( PQ \), and we are given the following:

- \( PT = 3x + 4 \)
- \( TQ = 5x - 6 \)

Since T is the midpoint, we know that \( PT = TQ \). Therefore, we can set up the equation:

\[
3x + 4 = 5x - 6
\]

Let's solve this equation step by step.

### Step 1: Move all terms involving \( x \) to one side of the equation.
\[
3x - 5x = -6 - 4
\]
\[
-2x = -10
\]

### Step 2: Solve for \( x \).
\[
x = \frac{-10}{-2} = 5
\]

### Step 3: Substitute \( x = 5 \) back into the expression for \( PT \).
\[
PT = 3(5) + 4 = 15 + 4 = 19
\]

Thus, \( PT = 19 \).

Would you like more details on the steps? Also, here are five questions to extend your understanding:

1. How would the solution change if T were not the midpoint?
2. Can you use this method to find \( PQ \) using \( PT \) and \( TQ \)?
3. What is the significance of midpoints in geometry?
4. How do we verify if our solution for \( x \) is correct?
5. What would happen if one of the expressions had a constant change?

**Tip**: Always double-check your work by substituting your solution back into the original equation to verify accuracy.

Use the figure to the right to find the value of PT. T is the midpoint of PQ. PT = 3x + 4 and TQ = 5x - 6.

Learn how to solve for PT using the midpoint theorem with the given expressions PT = 3x + 4 and TQ = 5x - 6. This problem involves key algebraic and geometric concepts and is suitable for students in Grades 8-10.

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