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To solve this problem, we need to find the point \((a, b)\) on the line \( 2x - 3y + 6 = 0 \) that is closest to the point \((3, 1)\). Once we have the coordinates of the closest point \((a, b)\), we will calculate \(a + b\).

### Step 1: Line Equation
The given line equation is:
\[
2x - 3y + 6 = 0
\]
This can be rewritten as:
\[
y = \frac{2}{3}x + 2
\]
The slope of the line is \(\frac{2}{3}\).

### Step 2: Perpendicular Line
The shortest distance from a point to a line is along the line perpendicular to the given line. The slope of the line perpendicular to the given line is the negative reciprocal of \(\frac{2}{3}\), which is \(-\frac{3}{2}\).

Using the point-slope form, we can write the equation of the line passing through \((3, 1)\) with slope \(-\frac{3}{2}\):
\[
y - 1 = -\frac{3}{2}(x - 3)
\]
Simplifying this:
\[
y - 1 = -\frac{3}{2}x + \frac{9}{2}
\]
\[
y = -\frac{3}{2}x + \frac{9}{2} + 1
\]
\[
y = -\frac{3}{2}x + \frac{11}{2}
\]

### Step 3: Find Intersection of the Two Lines
Now, we solve for the intersection of the line \( 2x - 3y + 6 = 0 \) and the perpendicular line we just found, \( y = -\frac{3}{2}x + \frac{11}{2} \).

Substitute \( y = -\frac{3}{2}x + \frac{11}{2} \) into \( 2x - 3y + 6 = 0 \):
\[
2x - 3\left(-\frac{3}{2}x + \frac{11}{2}\right) + 6 = 0
\]
Simplifying:
\[
2x + \frac{9}{2}x - \frac{33}{2} + 6 = 0
\]
Multiply everything by 2 to eliminate the fractions:
\[
4x + 9x - 33 + 12 = 0
\]
\[
13x - 21 = 0
\]
\[
x = \frac{21}{13}
\]

### Step 4: Find the Corresponding \(y\)-coordinate
Now substitute \(x = \frac{21}{13}\) into the equation of the perpendicular line:
\[
y = -\frac{3}{2}\left(\frac{21}{13}\right) + \frac{11}{2}
\]
\[
y = -\frac{63}{26} + \frac{143}{26} = \frac{80}{26} = \frac{40}{13}
\]

### Step 5: Calculate \(a + b\)
So, the closest point \((a, b)\) is \(\left(\frac{21}{13}, \frac{40}{13}\right)\).

Now, we can calculate \(a + b\):
\[
a + b = \frac{21}{13} + \frac{40}{13} = \frac{61}{13}
\]

Thus, \(a + b = \frac{61}{13}\).

Would you like more details or have any questions?

### Related Questions:
1. How do we derive the distance from a point to a line?
2. What is the geometric significance of perpendicular lines in finding shortest distances?
3. Could we use vector projections for this problem? How?
4. What happens when the point is on the line itself? What would the closest point be?
5. Can this method be generalized to 3D geometry?

### Tip:
When solving for the shortest distance from a point to a line, always remember that the path is along the perpendicular direction!

Find the sum point (a, b) on the line 2x - 3y + 6 = 0 that is closest to the point (3, 1). Calculate a + b.

This problem finds the closest point on the line 2x - 3y + 6 = 0 to the point (3, 1) using perpendicular distance methods. Includes steps for calculating the coordinates of the point and the sum a + b. Suitable for high school geometry students.

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