To find the equation of a line that is parallel to a given line and passes through a specific point, we can follow these steps:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = \frac{1}{5}x + 4 \)[/tex]. The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. Therefore, the slope ( [tex]\( m \)[/tex] ) of the given line is [tex]\( \frac{1}{5} \)[/tex].
2. Parallel lines have the same slope:
Since parallel lines have the same slope, the slope of the line we want to find will also be [tex]\( \frac{1}{5} \)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
Given the point [tex]\( (-2, 2) \)[/tex] and the slope [tex]\( \frac{1}{5} \)[/tex]:
[tex]\[
y - 2 = \frac{1}{5}(x + 2)
\][/tex]
4. Simplify the equation to slope-intercept form ( [tex]\( y = mx + b \)[/tex]):
[tex]\[
y - 2 = \frac{1}{5}(x + 2)
\][/tex]
Distribute [tex]\(\frac{1}{5}\)[/tex] on the right-hand side:
[tex]\[
y - 2 = \frac{1}{5}x + \frac{2}{5}
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{5}x + \frac{2}{5} + 2
\][/tex]
6. Combine like terms:
[tex]\[
y = \frac{1}{5}x + \frac{2}{5} + \frac{10}{5}
\][/tex]
7. Simplify the expression:
[tex]\[
y = \frac{1}{5}x + \frac{2 + 10}{5}
\][/tex]
[tex]\[
y = \frac{1}{5}x + \frac{12}{5}
\][/tex]
Therefore, the equation of the line that is parallel to [tex]\( y = \frac{1}{5}x + 4 \)[/tex] and passes through the point [tex]\( (-2, 2) \)[/tex] is [tex]\( y = \frac{1}{5}x + \frac{12}{5} \)[/tex].
Therefore, the correct choice is:
[tex]\[ y = \frac{1}{5}x + \frac{12}{5} \][/tex]
