To determine the equation of a line that is parallel to the given line [tex]\(y = \frac{1}{5} x + 4\)[/tex] and passes through the point [tex]\((-2, 2)\)[/tex], follow these steps:
1. Identify the Slope:
- The given line [tex]\(y = \frac{1}{5} x + 4\)[/tex] has a slope of [tex]\(\frac{1}{5}\)[/tex].
- Since parallel lines have the same slope, the new line we are seeking will also have a slope of [tex]\(\frac{1}{5}\)[/tex].
2. Use the Point-Slope Form of a Line:
- The point-slope form of a line 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 a point on the line.
- Given:
- Slope [tex]\(m = \frac{1}{5}\)[/tex]
- Point [tex]\((x_1, y_1) = (-2, 2)\)[/tex]
3. Substitute the Values into the Point-Slope Form:
- Substitute [tex]\(m = \frac{1}{5}\)[/tex], [tex]\(x_1 = -2\)[/tex], and [tex]\(y_1 = 2\)[/tex] into the point-slope form:
[tex]\[ y - 2 = \frac{1}{5}(x - (-2)) \][/tex]
[tex]\[ y - 2 = \frac{1}{5}(x + 2) \][/tex]
4. Simplify the Equation:
- Distribute [tex]\(\frac{1}{5}\)[/tex] through the parentheses:
[tex]\[ y - 2 = \frac{1}{5}x + \frac{1}{5}(2) \][/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]
- Convert 2 to a fraction with a common denominator of 5:
[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 [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]
