To find the equation of the line that is perpendicular to the given line [tex]\(6x - 2y = 8\)[/tex] and passes through the point [tex]\((-9, -2)\)[/tex], follow these steps:
1. Rewrite the given line in slope-intercept form: To determine the slope of the line, we need to express the given equation in the form [tex]\(y = mx + b\)[/tex].
Start with the equation:
[tex]\[
6x - 2y = 8
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
-2y = -6x + 8
\][/tex]
[tex]\[
y = 3x - 4
\][/tex]
Therefore, the slope ([tex]\(m_1\)[/tex]) of the given line is 3.
2. Determine the slope of the perpendicular line: The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope.
Given that the slope ([tex]\(m_1\)[/tex]) of the original line is 3, the slope ([tex]\(m_2\)[/tex]) of the perpendicular line is:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{3}
\][/tex]
3. Use the point-slope form to find the 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 and [tex]\(m\)[/tex] is the slope of the line.
Here, the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((-9, -2)\)[/tex] and the slope [tex]\(m_2 = -\frac{1}{3}\)[/tex]:
Substituting these values into the point-slope form equation:
[tex]\[
y - (-2) = -\frac{1}{3}(x - (-9))
\][/tex]
[tex]\[
y + 2 = -\frac{1}{3}(x + 9)
\][/tex]
4. Final equation: The equation of the line perpendicular to [tex]\(6x - 2y = 8\)[/tex] and passing through [tex]\((-9, -2)\)[/tex] in point-slope form is:
[tex]\[
y + 2 = -\frac{1}{3}(x + 9)
\][/tex]
In summary, the slope of the perpendicular line is [tex]\(-\frac{1}{3}\)[/tex] and the equation in point-slope form is:
[tex]\[
y + 2 = -\frac{1}{3}(x + 9)
\][/tex]
