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. Determine the slope of the given line:
The equation is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope. For the given line [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(-\frac{2}{3}\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the perpendicular slope [tex]\( m_{\perp} \)[/tex] will be:
[tex]\[
m_{\perp} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2}
\][/tex]
3. Use the point-slope form to determine the equation of the new line:
We need the equation of the line that passes through the point [tex]\((-2, -2)\)[/tex] with slope [tex]\( \frac{3}{2} \)[/tex]. The point-slope form is:
[tex]\[
y - y_1 = m_{\perp}(x - x_1)
\][/tex]
Plugging in the point [tex]\((-2, -2)\)[/tex] and the slope [tex]\( \frac{3}{2} \)[/tex], we get:
[tex]\[
y - (-2) = \frac{3}{2}(x - (-2))
\][/tex]
Simplify this:
[tex]\[
y + 2 = \frac{3}{2}(x + 2)
\][/tex]
4. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
Distribute [tex]\( \frac{3}{2} \)[/tex] on the right-hand side:
[tex]\[
y + 2 = \frac{3}{2}x + \frac{3}{2} \cdot 2
\][/tex]
[tex]\[
y + 2 = \frac{3}{2}x + 3
\][/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 in slope-intercept form 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]
The correct choice is:
[tex]\[
\boxed{y = \frac{3}{2} x + 1}
\][/tex]
