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To find the point on the line \( y = 2x - 3 \) that is closest to the point \( (5, 1) \), we need to find the perpendicular distance from the point \( (5, 1) \) to the line.

### Step-by-Step Solution:

1. **Slope of the given line**:  
   The equation of the line is \( y = 2x - 3 \), which is in slope-intercept form \( y = mx + b \), where the slope \( m = 2 \).

2. **Slope of the perpendicular line**:  
   The slope of a line perpendicular to \( y = 2x - 3 \) is the negative reciprocal of 2. So, the slope of the perpendicular line is \( m_{\text{perp}} = -\frac{1}{2} \).

3. **Equation of the perpendicular line**:  
   We can now write the equation of the line that passes through the point \( (5, 1) \) and has a slope of \( -\frac{1}{2} \). Using the point-slope form \( y - y_1 = m(x - x_1) \), we get:
   \[
   y - 1 = -\frac{1}{2}(x - 5)
   \]
   Simplifying:
   \[
   y - 1 = -\frac{1}{2}x + \frac{5}{2}
   \]
   \[
   y = -\frac{1}{2}x + \frac{7}{2}
   \]

4. **Find the point of intersection**:  
   Now, we find the point where the perpendicular line intersects the original line \( y = 2x - 3 \). Set the equations equal to each other:
   \[
   2x - 3 = -\frac{1}{2}x + \frac{7}{2}
   \]
   Multiply through by 2 to eliminate fractions:
   \[
   4x - 6 = -x + 7
   \]
   Solve for \( x \):
   \[
   5x = 13
   \]
   \[
   x = \frac{13}{5}
   \]

5. **Find the corresponding \( y \)-coordinate**:  
   Substitute \( x = \frac{13}{5} \) into the equation of the original line \( y = 2x - 3 \):
   \[
   y = 2\left(\frac{13}{5}\right) - 3 = \frac{26}{5} - 3 = \frac{26}{5} - \frac{15}{5} = \frac{11}{5}
   \]

6. **Conclusion**:  
   The point on the line \( y = 2x - 3 \) that is closest to \( (5, 1) \) is \( \left( \frac{13}{5}, \frac{11}{5} \right) \).

Would you like any additional details or explanations?

Here are some related questions to explore further:

1. How do you derive the formula for the distance between a point and a line?
2. Can this method be generalized to three-dimensional geometry?
3. What is the geometric interpretation of the negative reciprocal in perpendicular slopes?
4. How can you verify that this point is indeed the closest one?
5. How would the process change if the equation of the line were given in standard form?

**Tip**: When dealing with perpendicular distances, always remember that the slopes of perpendicular lines are negative reciprocals of each other.

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 point on the line y = 2x - 3 that is closest to the point (5, 1). This step-by-step solution involves calculating the slope, using the point-slope form, and finding the intersection. Suitable for high school students learning about slopes and perpendicular lines.

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