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The problem asks you to write the equation of the line passing through the point \( P(0, 5) \) and having the same slope (coefficient angolare) as the line with the equation \( 2x - 3y + 1 = 0 \).

### Step-by-step solution:

1. **Find the slope of the given line**:
   The line equation provided is \( 2x - 3y + 1 = 0 \). Let's rewrite it in the slope-intercept form \( y = mx + b \), where \( m \) is the slope.

   \[
   2x - 3y + 1 = 0 \quad \text{(Move terms around)}
   \]
   \[
   -3y = -2x - 1 \quad \text{(Isolate \( y \))}
   \]
   \[
   y = \frac{2}{3}x + \frac{1}{3}
   \]

   From this, the slope \( m \) of the line is \( \frac{2}{3} \).

2. **Write the equation of the new line**:
   We need to find the equation of the line passing through \( P(0, 5) \) with the same slope \( m = \frac{2}{3} \). Using the point-slope form of the line equation:
   \[
   y - y_1 = m(x - x_1)
   \]
   where \( (x_1, y_1) \) is the point \( P(0, 5) \) and \( m = \frac{2}{3} \).
   \[
   y - 5 = \frac{2}{3}(x - 0)
   \]
   \[
   y - 5 = \frac{2}{3}x
   \]
   \[
   y = \frac{2}{3}x + 5
   \]

Thus, the equation of the line is:
\[
y = \frac{2}{3}x + 5
\]

This is the desired equation.

Do you need further clarifications or details? Here are five related questions you might find helpful:

1. How do you find the slope of a line from its general equation?
2. What is the point-slope form of a line, and when is it useful?
3. How can you derive the slope-intercept form from the point-slope form?
4. What does the slope of a line represent geometrically?
5. How would the equation change if the line passed through a different point?

**Tip:** Always rewrite the equation in slope-intercept form to easily identify the slope and intercepts of the line.

Scrivi l'equazione della retta che passa per il punto P(0; 5) e ha coefficiente angolare uguale a quello della retta di equazione 2x - 3y + 1 = 0.

Learn how to write the equation of a line passing through point P(0,5) with the same slope as the line 2x - 3y + 1 = 0. This problem covers linear equations, slope-intercept form, and point-slope form. Suitable for students in Grades 8-10.

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