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We need to find the distance between the point \( (2, 5) \) and a line passing through \( (-2, 3) \) that is parallel to the line \( 2y = 6x - 5 \).

### Step 1: Simplify the given line equation.
The equation \( 2y = 6x - 5 \) simplifies to:
\[
y = 3x - \frac{5}{2}
\]
The slope of this line is \( m = 3 \).

### Step 2: Equation of the parallel line.
The line we seek is parallel to this line, so it has the same slope \( m = 3 \). Using the point-slope form, the equation of the line passing through \( (-2, 3) \) is:
\[
y - 3 = 3(x + 2)
\]
Simplifying:
\[
y = 3x + 9
\]

### Step 3: Distance formula.
The formula to find the distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is:
\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]
We rewrite \( y = 3x + 9 \) as \( 3x - y + 9 = 0 \), so \( A = 3 \), \( B = -1 \), and \( C = 9 \).

Substitute \( (x_1, y_1) = (2, 5) \) into the formula:
\[
d = \frac{|3(2) - 1(5) + 9|}{\sqrt{3^2 + (-1)^2}} = \frac{|6 - 5 + 9|}{\sqrt{9 + 1}} = \frac{|10|}{\sqrt{10}} = \frac{10}{\sqrt{10}} = \sqrt{10}
\]

So, the distance is \( \sqrt{10} \).

Would you like further clarification or details?

Here are 5 follow-up questions you might find helpful:
1. How is the point-slope form of a line derived?
2. Why do parallel lines have the same slope?
3. What are the geometric properties of lines with equal slopes?
4. Can the distance formula for a point and line be derived geometrically?
5. How does changing the slope affect the distance between a point and a line?

**Tip:** Understanding the point-slope form can help you quickly write equations for lines parallel to any given line.

Find the distance of the point (2,5) from the line passing through (-2,3) and parallel to the line whose equation is 2y = 6x - 5.

This problem calculates the distance of the point (2,5) from a line passing through (-2,3) and parallel to the line 2y = 6x - 5. It involves the concepts of parallel lines, the point-slope form of a line, and the distance formula.

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