To find the equation of the line that is parallel to a given line and passes through a specific point, we need to follow these steps:
1. Identify the slope of the given line.
The given line equation is [tex]\( y = \frac{1}{5}x + 4 \)[/tex].
The slope ([tex]\( m \)[/tex]) of this line is [tex]\( \frac{1}{5} \)[/tex].
2. Since parallel lines have the same slope, the slope of the new line will be the same:
[tex]\[ m = \frac{1}{5} \][/tex]
3. Use the point-slope form of the equation of a line to find the new line's equation.
The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, the point through which the line passes is [tex]\((-2, 2)\)[/tex]. Thus, [tex]\( x_1 = -2 \)[/tex] and [tex]\( y_1 = 2 \)[/tex].
4. Substitute the slope and the point into the point-slope form:
[tex]\[ y - 2 = \frac{1}{5}(x + 2) \][/tex]
5. Simplify this equation to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 2 = \frac{1}{5}x + \frac{1}{5}(2) \][/tex]
[tex]\[ y - 2 = \frac{1}{5}x + \frac{2}{5} \][/tex]
[tex]\[ y = \frac{1}{5}x + \frac{2}{5} + 2 \][/tex]
[tex]\[ y = \frac{1}{5}x + \frac{2}{5} + \frac{10}{5} \][/tex]
[tex]\[ y = \frac{1}{5}x + \frac{12}{5} \][/tex]
So, the equation of the line that is parallel to the given line and passes through the point [tex]\((-2, 2)\)[/tex] is:
[tex]\[ y = \frac{1}{5}x + \frac{12}{5} \][/tex]
Therefore, the correct equation among the given choices is [tex]\( \boxed{y = \frac{1}{5} x + \frac{12}{5}} \)[/tex].
