To find the equation of a line that is parallel to a given line and passes through a specific point, we will follow these steps:
1. Identify the slope of the given line:
The given line has the equation [tex]\( y = \frac{1}{5}x + 4 \)[/tex]. The slope of this line, [tex]\( m \)[/tex], is the coefficient of [tex]\( x \)[/tex], which is [tex]\( \frac{1}{5} \)[/tex].
2. Determine the slope of the parallel line:
Parallel lines have the same slope. Therefore, the slope of the line we need to find is also [tex]\( \frac{1}{5} \)[/tex].
3. Use the point-slope form of the line equation:
The point-slope form of a line's equation is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( (x_1, y_1) \)[/tex] is a point on the line.
Here, we are given the point [tex]\((-2, 2)\)[/tex] and the slope [tex]\( m = \frac{1}{5} \)[/tex].
So, we substitute these values into the point-slope form:
[tex]\[
y - 2 = \frac{1}{5}(x + 2)
\][/tex]
4. Solve for [tex]\( y \)[/tex] to get the slope-intercept form:
Simplify the equation:
[tex]\[
y - 2 = \frac{1}{5}x + \frac{1}{5} \cdot 2
\][/tex]
[tex]\[
y - 2 = \frac{1}{5}x + \frac{2}{5}
\][/tex]
Add 2 to both sides to solve for [tex]\( y \)[/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]
5. Compare with the given choices:
We need to determine which of the provided choices matches [tex]\( y = \frac{1}{5}x + \frac{12}{5} \)[/tex]:
- [tex]\( y = \frac{1}{5}x + 4 \)[/tex]
- [tex]\( y = \frac{1}{5}x + \frac{12}{5} \)[/tex]
- [tex]\( y = -5x + 4 \)[/tex]
- [tex]\( y = -5x + \frac{12}{5} \)[/tex]
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].
Comparing this with the choices given, we find that the correct answer is:
[tex]\[ \boxed{2} \][/tex]
