To find the equation of the line that is perpendicular to the line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex], we need to follow these steps:
1. Find the slope of the given line:
The given line is in point-slope form: [tex]\( y - y_1 = m(x - x_1) \)[/tex].
From [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], we see that 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]\[
\text{Slope of perpendicular line} = -\frac{1}{m} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2}
\][/tex]
3. Use the point-slope form to write the equation of the perpendicular line:
We will use the point [tex]\((-2, -2)\)[/tex] and the slope [tex]\(\frac{3}{2}\)[/tex].
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plugging in the point and the slope:
[tex]\[
y - (-2) = \frac{3}{2}(x - (-2))
\][/tex]
Simplifying:
[tex]\[
y + 2 = \frac{3}{2}(x + 2)
\][/tex]
[tex]\[
y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2
\][/tex]
[tex]\[
y + 2 = \frac{3}{2}x + 3
\][/tex]
4. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3}{2}x + 3 - 2
\][/tex]
[tex]\[
y = \frac{3}{2}x + 1
\][/tex]
So, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[
y = \frac{3}{2}x + 1
\][/tex]
Hence, the correct answer is:
[tex]\[
y = \frac{3}{2}x + 1
\][/tex]
