The line y = 2x - 5 intersects the lines x = 5 and y = -2 at the points P and Q respectively. Find the midpoint of PQ.

We are given the line \( y = 2x - 5 \) and asked to find the midpoint of the segment formed by its intersection points with the lines \( x = 5 \) and \( y = -2 \). Let's break it down step by step.

### Step 1: Find the coordinates of point P
The line \( x = 5 \) is a vertical line, meaning the \( x \)-coordinate of the point of intersection is \( x = 5 \).

To find the \( y \)-coordinate of point \( P \), substitute \( x = 5 \) into the equation of the line \( y = 2x - 5 \):
\[
y = 2(5) - 5 = 10 - 5 = 5
\]
Thus, the coordinates of point \( P \) are \( (5, 5) \).

### Step 2: Find the coordinates of point Q
The line \( y = -2 \) is a horizontal line, meaning the \( y \)-coordinate of the point of intersection is \( y = -2 \).

To find the \( x \)-coordinate of point \( Q \), substitute \( y = -2 \) into the equation of the line \( y = 2x - 5 \):
\[
-2 = 2x - 5
\]
Solving for \( x \):
\[
2x = -2 + 5 = 3
\]
\[
x = \frac{3}{2}
\]
Thus, the coordinates of point \( Q \) are \( \left( \frac{3}{2}, -2 \right) \).

### Step 3: Find the midpoint of PQ
The formula for the midpoint \( M \) of a line segment connecting two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substitute the coordinates of \( P(5, 5) \) and \( Q \left( \frac{3}{2}, -2 \right) \) into the midpoint formula:
\[
M = \left( \frac{5 + \frac{3}{2}}{2}, \frac{5 + (-2)}{2} \right)
\]
Simplify the coordinates:
\[
M = \left( \frac{\frac{10}{2} + \frac{3}{2}}{2}, \frac{5 - 2}{2} \right) = \left( \frac{\frac{13}{2}}{2}, \frac{3}{2} \right) = \left( \frac{13}{4}, \frac{3}{2} \right)
\]

### Final Answer:
The midpoint of \( PQ \) is \( \left( \frac{13}{4}, \frac{3}{2} \right) \).

Would you like any more details or have any questions?

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Here are 5 related questions to extend this problem:
1. What is the distance between points \( P \) and \( Q \)?
2. How would you find the equation of the perpendicular bisector of line segment \( PQ \)?
3. What is the slope of the line segment connecting points \( P \) and \( Q \)?
4. How does the slope of the line segment \( PQ \) compare to the slope of the line \( y = 2x - 5 \)?
5. What is the area of the triangle formed by the points \( P \), \( Q \), and the origin?

**Tip:** When finding midpoints, always ensure both the x- and y-coordinates are averaged accurately to avoid mistakes in complex problems.

Find the midpoint of the line segment formed by the intersection of the line y = 2x - 5 with x = 5 and y = -2. A detailed step-by-step solution explains how to find the intersection points and use the midpoint formula, suitable for Grades 8-10 students.

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