Question12. Write an equation of the line that passes through the point $(4,-6)$ and is perpendicular to the line $y=-\frac{2}{3} x+1 .[4 p t s]$ $\begin{array}{l} \left(x_{1}, y_{1}\right)=(4,-6) \\ y=-\frac{2}{3} x+1-7 y=\frac{3}{2} x+1 \end{array}$ $y-y_{1}=m\left(x-x_{1}\right)$ $\begin{array}{l} y-(-6)=\frac{3}{2}(x-4) \\ y-(-6)=\frac{3 x}{2}+-4 \end{array}$

Studdy Solution

STEP 1

1. We need to find the equation of a line that passes through the point $(4, -6)$ .
2. The line is perpendicular to the line given by the equation $y = -\frac{2}{3}x + 1$ .
3. The slope of the line we are looking for is the negative reciprocal of $-\frac{2}{3}$ .

STEP 2

1. Determine the slope of the perpendicular line.
2. Use the point-slope form to write the equation of the line.
3. Simplify the equation to the desired form.

STEP 3

Determine the slope of the perpendicular line.
The slope of the given line is $-\frac{2}{3}$ . The slope of a line perpendicular to it is the negative reciprocal.
$m = -\left(-\frac{1}{\frac{2}{3}}\right) = \frac{3}{2}$

STEP 4

Use the point-slope form to write the equation of the line.
The point-slope form of a line is given by:
$y - y_1 = m(x - x_1)$
Substitute the point $(4, -6)$ and the slope $m = \frac{3}{2}$ :
$y - (-6) = \frac{3}{2}(x - 4)$

STEP 5

Simplify the equation to the desired form.
First, simplify the left side:
$y + 6 = \frac{3}{2}(x - 4)$
Distribute the $\frac{3}{2}$ :
$y + 6 = \frac{3}{2}x - \frac{3}{2} \times 4$
$y + 6 = \frac{3}{2}x - 6$
Subtract 6 from both sides to solve for $y$ :
$y = \frac{3}{2}x - 6 - 6$
$y = \frac{3}{2}x - 12$
The equation of the line is:
$y = \frac{3}{2}x - 12$

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