To find the equation of the line that is perpendicular to the given line and passes through the specified point, we need to follow these steps:
1. Identify the slope of the given line:
The given line is in point-slope form: [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex].
The slope (m) of the given line is [tex]\( -\frac{2}{3} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of [tex]\( -\frac{2}{3} \)[/tex] is [tex]\( \frac{3}{2} \)[/tex]. Hence, the slope of the perpendicular line is [tex]\( \frac{3}{2} \)[/tex].
3. Write the equation of the perpendicular line in slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
We know the slope [tex]\( m = \frac{3}{2} \)[/tex] and that it passes through the point [tex]\( (-2, -2) \)[/tex].
4. Determine the y-intercept (b) using the point [tex]\( (-2, -2) \)[/tex]:
Substitute [tex]\( m \)[/tex], [tex]\( x \)[/tex], and [tex]\( y \)[/tex] into the slope-intercept form equation to solve for [tex]\( b \)[/tex]:
[tex]\[
-2 = \frac{3}{2}(-2) + b
\][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[
-2 = -3 + b
\][/tex]
[tex]\[
b = 1
\][/tex]
5. Formulate the equation:
Using the slope and y-intercept, the equation of the perpendicular line is:
[tex]\[
y = \frac{3}{2} x + 1
\][/tex]
Thus, the equation of the line that is perpendicular to [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\( (-2, -2) \)[/tex] is [tex]\( y = \frac{3}{2} x + 1 \)[/tex].
The correct option is:
[tex]\[ y = \frac{3}{2} x + 1 \][/tex]
So, the correct choice from the given options is the fourth one. Thus, the result is 4.
