To find the equation of the line that passes through the point [tex]\((-7, 2)\)[/tex] and is parallel to the given line [tex]\(y = \frac{2}{5} x - \frac{1}{2}\)[/tex], follow these steps:
1. Identify the slope: Since we need a parallel line, it will have the same slope as the given line. The slope of the given line [tex]\(y = \frac{2}{5} x - \frac{1}{2}\)[/tex] is [tex]\(\frac{2}{5}\)[/tex].
2. Use the point-slope form: The equation of a line in point-slope form is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the point the line passes through and [tex]\(m\)[/tex] is the slope.
3. Substitute the known values: The point given is [tex]\((-7, 2)\)[/tex] and the slope [tex]\(m\)[/tex] is [tex]\(\frac{2}{5}\)[/tex]. Substituting these into the point-slope form gives:
[tex]\[
y - 2 = \frac{2}{5}(x - (-7))
\][/tex]
4. Simplify the equation:
[tex]\[
y - 2 = \frac{2}{5}(x + 7)
\][/tex]
5. Distribute the slope:
[tex]\[
y - 2 = \frac{2}{5} x + \frac{2}{5} \times 7
\][/tex]
[tex]\[
y - 2 = \frac{2}{5} x + \frac{14}{5}
\][/tex]
6. Isolate [tex]\(y\)[/tex]: Add 2 to both sides of the equation. Remember to convert 2 into a fraction with a common denominator:
[tex]\[
y = \frac{2}{5} x + \frac{14}{5} + 2
\][/tex]
[tex]\[
y = \frac{2}{5} x + \frac{14}{5} + \frac{10}{5}
\][/tex]
[tex]\[
y = \frac{2}{5} x + \frac{24}{5}
\][/tex]
So, the equation of the line that passes through the point [tex]\((-7, 2)\)[/tex] and is parallel to the given line is:
[tex]\[
y = \frac{2}{5} x + \frac{24}{5}
\][/tex]
