To find the equation of a line that is perpendicular to the given line and passes through a specific point, follow these steps:
### Step 1: Convert the given line to slope-intercept form
The given line is [tex]\( y - 4 = \frac{2}{3}(x - 6) \)[/tex].
First, let’s distribute the [tex]\(\frac{2}{3}\)[/tex] on the right side:
[tex]\[ y - 4 = \frac{2}{3} x - \frac{2}{3} \cdot 6 \][/tex]
[tex]\[ y - 4 = \frac{2}{3} x - 4 \][/tex]
Next, we add 4 to both sides of the equation to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{2}{3} x - 4 + 4 \][/tex]
[tex]\[ y = \frac{2}{3} x \][/tex]
So, the equation of the given line in slope-intercept form is:
[tex]\[ y = \frac{2}{3} x - 4 \][/tex]
The slope ([tex]\( m_1 \)[/tex]) of this line is [tex]\(\frac{2}{3}\)[/tex].
### Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the slope ([tex]\( m_2 \)[/tex]) of the perpendicular line to [tex]\( \frac{2}{3} \)[/tex] is:
[tex]\[ m_2 = -\frac{1}{(\frac{2}{3})} = -\frac{3}{2} \][/tex]
### Step 3: Use the point-slope form to find the equation of the perpendicular line
We know the perpendicular line passes through the point [tex]\((-2, -2)\)[/tex], and its slope is [tex]\(-\frac{3}{2}\)[/tex]. We can use the point-slope form of a line equation, which is:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
Substitute [tex]\( x_1 = -2 \)[/tex], [tex]\( y_1 = -2 \)[/tex], and [tex]\( m = -\frac{3}{2} \)[/tex]:
[tex]\[ y - (-2) = -\frac{3}{2} (x - (-2)) \][/tex]
[tex]\[ y + 2 = -\frac{3}{2} (x + 2) \][/tex]
### Step 4: Simplify to slope-intercept form
Distribute the [tex]\(-\frac{3}{2}\)[/tex] on the right 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 solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{3}{2} x - 3 - 2 \][/tex]
[tex]\[ y = -\frac{3}{2} x - 5 \][/tex]
Thus, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[ y = -\frac{3}{2} x - 5 \][/tex]
This matches none of the provided options directly. Let's revise provided options as,[tex]\(y=-\frac{3}{2} x - 5 \)[/tex],is simplified form of [tex]\( y = -1.5x - 5 \)[/tex], then from this we can chose correct option.
So, none of the provided options should be correct [tex]\( y=-\frac{3}{2} x - 5 \)[/tex], this satisfies all above calculations statements.
