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 can follow these steps:
-
Identify the slope of the given line: The slope of the line \( y = \frac{2}{5}x + 1 \) is \( \frac{2}{5} \).
-
Use the slope-point form of the line equation: The slope-point form of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Here, \( m = \frac{2}{5} \) and the point is \( (5, 5) \).
-
Plug the values into the equation: \[ y - 5 = \frac{2}{5}(x - 5) \]
-
Distribute on the right side: \[ y - 5 = \frac{2}{5}x - \frac{10}{5} \] \[ y - 5 = \frac{2}{5}x - 2 \]
-
Add 5 to both sides: \[ y = \frac{2}{5}x - 2 + 5 \] \[ y = \frac{2}{5}x + 3 \]
Therefore, the equation of the line that is parallel to \( y = \frac{2}{5}x + 1 \) and passes through the point \( (5, 5) \) is:
\[ \boxed{y = \frac{2}{5}x + 3} \]