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], we need to follow these steps:
1. Identify the slope of the given line and determine the perpendicular slope:
- The given equation is in point-slope form: [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex].
- The slope [tex]\( m \)[/tex] of this line is [tex]\( -\frac{2}{3} \)[/tex].
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- Therefore, the slope of the perpendicular line is the negative reciprocal of [tex]\( -\frac{2}{3} \)[/tex], which is [tex]\( \frac{3}{2} \)[/tex].
2. Use the point-slope form of the line equation with the perpendicular slope and the given point:
- The point-slope form of a line equation is given by [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is a point on the line.
- Substituting the values, we get:
[tex]\[
y - (-2) = \frac{3}{2}(x - (-2))
\][/tex]
- Simplify this to:
[tex]\[
y + 2 = \frac{3}{2}(x + 2)
\][/tex]
3. Distribute the slope on the right-hand side:
- Distribute [tex]\( \frac{3}{2} \)[/tex] to both terms inside the parentheses:
[tex]\[
y + 2 = \frac{3}{2} x + \frac{3}{2} \cdot 2
\][/tex]
- Simplifying further:
[tex]\[
y + 2 = \frac{3}{2} x + 3
\][/tex]
4. Solve for [tex]\( y \)[/tex] to put the equation in slope-intercept form:
- Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3}{2} x + 3 - 2
\][/tex]
- Simplify the right-hand side:
[tex]\[
y = \frac{3}{2} x + 1
\][/tex]
5. Result:
- The equation of the line in slope-intercept form is [tex]\( y = \frac{3}{2} x + 1 \)[/tex].
Therefore, the correct equation of the line perpendicular to [tex]\( y - 4 = -\frac{2}{3}(x - 6) \)[/tex] and passing through [tex]\((-2, -2)\)[/tex] is:
[tex]\[
y = \frac{3}{2} x + 1
\][/tex]
Among the given choices, the correct answer is:
[tex]\[ y = \frac{3}{2} x + 1 \][/tex]
