To find the equation of a line that is parallel to a given line and passes through a specific point, we use the fact that parallel lines have the same slope.
Let’s consider the given line:
[tex]\[ y = \frac{1}{3}x + 4. \][/tex]
The slope of this line is [tex]\(\frac{1}{3}\)[/tex]. Since parallel lines share the same slope, the slope of our new line will also be [tex]\(\frac{1}{3}\)[/tex].
Next, we'll use the point-slope form of the equation of a line, which is given by:
[tex]\[ y - y_1 = m(x - x_1), \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point through which the line passes and [tex]\(m\)[/tex] is the slope of the line.
Here, we want the line to pass through the point [tex]\((-2, 2)\)[/tex] and have a slope [tex]\(m = \frac{1}{3}\)[/tex]. Plugging in the values, we get:
[tex]\[ y - 2 = \frac{1}{3}(x - (-2)). \][/tex]
Simplify the equation:
[tex]\[ y - 2 = \frac{1}{3}(x + 2). \][/tex]
Distribute the [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ y - 2 = \frac{1}{3}x + \frac{2}{3}. \][/tex]
To find the y-intercept, solve for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{1}{3}x + \frac{2}{3} + 2. \][/tex]
Convert 2 to a fraction with a common denominator:
[tex]\[ y = \frac{1}{3}x + \frac{2}{3} + \frac{6}{3}. \][/tex]
Combine the terms:
[tex]\[ y = \frac{1}{3}x + \frac{8}{3}. \][/tex]
Thus, the equation of the line that is parallel to the given line and passes through the point [tex]\((-2,2)\)[/tex] is:
[tex]\[ y = \frac{1}{3}x + \frac{8}{3}. \][/tex]
