To determine the equation of the line that is perpendicular to the given line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex], let's follow these steps:
1. Convert the given line equation to slope-intercept form:
The given line is written in point-slope form:
[tex]\[
y - 4 = -\frac{2}{3}(x - 6)
\][/tex]
Simplify it to get the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 4 = -\frac{2}{3}x + 4 \quad \Rightarrow \quad y = -\frac{2}{3}x + 4 + 4 \quad \Rightarrow \quad y = -\frac{2}{3}x + 8
\][/tex]
Therefore, the slope [tex]\( m \)[/tex] of the given line is [tex]\(-\frac{2}{3}\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
[tex]\[
\text{Slope of the original line} = -\frac{2}{3} \quad \Rightarrow \quad \text{Slope of the perpendicular line} = \frac{3}{2}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
We know the perpendicular line passes through the point [tex]\((-2, -2)\)[/tex] and has a slope of [tex]\(\frac{3}{2}\)[/tex]. The point-slope form of a line equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting the known values [tex]\(x_1 = -2, y_1 = -2, m = \frac{3}{2}\)[/tex]:
[tex]\[
y - (-2) = \frac{3}{2}(x - (-2)) \quad \Rightarrow \quad y + 2 = \frac{3}{2}(x + 2)
\][/tex]
Simplify:
[tex]\[
y + 2 = \frac{3}{2}x + 3 \quad \Rightarrow \quad y = \frac{3}{2}x + 3 - 2 \quad \Rightarrow \quad y = \frac{3}{2}x + 1
\][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through [tex]\((-2, -2)\)[/tex] is:
[tex]\[
y = \frac{3}{2} x + 1
\][/tex]
The correct choice is:
[tex]\[
\boxed{y = \frac{3}{2} x + 1}
\][/tex]
