To find 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. Identify the slope of the given line:
The given line equation is in point-slope form:
[tex]\[
y - 4 = -\frac{2}{3}(x - 6)
\][/tex]
The coefficient of [tex]\((x - 6)\)[/tex] is the slope of the line. So, the slope [tex]\( m_1 \)[/tex] of the given line is:
[tex]\[
m_1 = -\frac{2}{3}
\][/tex]
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope [tex]\( m_2 \)[/tex] of the perpendicular line is:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2}
\][/tex]
3. Use the slope-intercept form (y = mx + b) to find the y-intercept:
The point [tex]\((-2, -2)\)[/tex] lies on the perpendicular line. Using this point and the slope [tex]\( \frac{3}{2} \)[/tex], we can find the y-intercept [tex]\( b \)[/tex]:
[tex]\[
y = mx + b
\][/tex]
Substituting [tex]\((x_1, y_1) = (-2, -2)\)[/tex] and [tex]\( m_2 = \frac{3}{2} \)[/tex]:
[tex]\[
-2 = \frac{3}{2}(-2) + b
\][/tex]
4. Solve for the y-intercept [tex]\( b \)[/tex]:
Calculate [tex]\( \frac{3}{2}(-2) \)[/tex]:
[tex]\[
\frac{3}{2} \times -2 = -3
\][/tex]
Substitute back into the equation to solve for [tex]\( b \)[/tex]:
[tex]\[
-2 = -3 + b
\][/tex]
Adding 3 to both sides:
[tex]\[
b = 1
\][/tex]
5. Write the final equation:
With [tex]\( m_2 = \frac{3}{2} \)[/tex] and [tex]\( b = 1 \)[/tex], the equation of the line in slope-intercept form is:
[tex]\[
y = \frac{3}{2}x + 1
\][/tex]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[
\boxed{y = \frac{3}{2}x + 1}
\][/tex]
