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:
Since we are looking for a parallel line, it will have the same slope as the given line. Here, the given slope is [tex]\(\frac{1}{5}\)[/tex].
2. Use the point-slope form of the line equation:
The point-slope form is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point given. In this case, the point is [tex]\((-2, 2)\)[/tex].
3. Substitute the slope and point into the point-slope form:
[tex]\[
y - 2 = \frac{1}{5}(x - (-2))
\][/tex]
4. Simplify the equation:
[tex]\[
y - 2 = \frac{1}{5}(x + 2)
\][/tex]
First distribute the slope:
[tex]\[
y - 2 = \frac{1}{5}x + \frac{1}{5} \cdot 2
\][/tex]
[tex]\[
y - 2 = \frac{1}{5}x + \frac{2}{5}
\][/tex]
5. Isolate y to convert the equation to slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[
y = \frac{1}{5}x + \frac{2}{5} + 2
\][/tex]
6. Add the constant term on the right side:
[tex]\[
y = \frac{1}{5}x + \frac{2}{5} + \frac{10}{5}
\][/tex]
[tex]\[
y = \frac{1}{5}x + \frac{12}{5}
\][/tex]
Therefore, 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]
Hence, the correct option is:
[tex]\[
y = \frac{1}{5}x + \frac{12}{5}
\][/tex]
