To determine the equation of a line that is parallel to the given line and passes through the specified point, we follow these steps:
1. Identify the slope of the given line: The given line is [tex]\( y = \frac{1}{5} x + 4 \)[/tex]. The slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{1}{5} \)[/tex].
2. Remember the property of parallel lines: If two lines are parallel, their slopes are equal. Hence, the slope of the line we are looking for is also [tex]\( \frac{1}{5} \)[/tex].
3. Use the point-slope form of the equation of a line: The point-slope form 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. Here, the point is [tex]\( (-2, 2) \)[/tex] and the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].
4. Substitute the values into the point-slope form:
[tex]\[
y - 2 = \frac{1}{5}(x - (-2))
\][/tex]
5. Simplify the equation:
[tex]\[
y - 2 = \frac{1}{5}(x + 2)
\][/tex]
[tex]\[
y - 2 = \frac{1}{5}x + \frac{1}{5} \cdot 2
\][/tex]
[tex]\[
y - 2 = \frac{1}{5}x + \frac{2}{5}
\][/tex]
6. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{5}x + \frac{2}{5} + 2
\][/tex]
Convert the constant term to a common denominator:
[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].
Among the provided options, the correct one is:
[tex]\[ y = \frac{1}{5} x + \frac{12}{5} \][/tex]
