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The problem asks us to find the point on the line \( y = 2x - 3 \) that is closest to the point \( (5, 1) \) in the xy-plane. 

### Approach:

To find the point on the line that is closest to \( (5, 1) \), we follow these steps:

1. **Find the slope of the perpendicular line**: The slope of the given line is 2. The slope of a line perpendicular to this will be the negative reciprocal, which is \( -\frac{1}{2} \).

2. **Write the equation of the perpendicular line**: Using the point \( (5, 1) \), we can form the equation of the line that passes through \( (5, 1) \) and has a slope of \( -\frac{1}{2} \). The equation of the line will be in the point-slope form:
   \[
   y - 1 = -\frac{1}{2}(x - 5)
   \]

3. **Find the intersection of the two lines**: Solve the system of equations:
   - \( y = 2x - 3 \) (the original line)
   - \( y = -\frac{1}{2}x + \frac{7}{2} \) (the perpendicular line)

4. **Find the coordinates of the intersection**: This will give us the point on the original line that is closest to \( (5, 1) \).

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Here are five questions related to this type of problem:

1. How do you find the distance between a point and a line in the coordinate plane?
2. What is the significance of perpendicular lines in optimization problems involving distances?
3. Can you find the distance from the point \( (5, 1) \) to the line \( y = 2x - 3 \)?
4. How does the method of solving systems of linear equations help in geometry problems?
5. What other methods can be used to minimize the distance between a point and a line?

**Tip**: When solving distance problems, remember that the shortest distance between a point and a line is along the perpendicular line.

Find the point on the line y = 2x - 3 in the xy-plane that is closest to the point (5, 1).

Learn how to find the closest point on the line y = 2x - 3 to the point (5, 1) using perpendicular slopes and solving a system of equations. Suitable for high school students studying coordinate geometry.

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