To find the equation of a line that is parallel to the line \( y = \frac{2}{5}x + 1 \) and passes through the point \( (5, 5) \), we first note that parallel lines have the same slope.
The slope of the given line is \( \frac{2}{5} \). Therefore, the line we seek will also have this slope.
Next, we can use the point-slope form of a line's equation, which is given by:
\[ y - y_1 = m(x - x_1) \]
Here, \( m \) is the slope, and \( (x_1, y_1) \) is the point the line passes through. For our line:
- \( m = \frac{2}{5} \)
- \( (x_1, y_1) = (5, 5) \)
Plugging these values into the point-slope form, we have:
\[ y - 5 = \frac{2}{5}(x - 5) \]
Now, we'll simplify this equation:
- Distribute the slope on the right side:
\[ y - 5 = \frac{2}{5}x - \frac{2}{5} \cdot 5 \]
This simplifies to:
\[ y - 5 = \frac{2}{5}x - 2 \]
- Add 5 to both sides to solve for \( y \):
\[ y = \frac{2}{5}x - 2 + 5 \]
This simplifies to:
\[ y = \frac{2}{5}x + 3 \]
Thus, the equation of the line that is parallel to \( y = \frac{2}{5}x + 1 \) and passes through the point \( (5, 5) \) is:
\[ y = \frac{2}{5}x + 3 \]