QuestionFind the line equation through $(-4,1)$ that is perpendicular to $-3x + 4y = 9$ .

Studdy Solution

STEP 1

Assumptions1. The line passes through the point $(-4,1)$ . The line is perpendicular to the line with the equation $-3x +4y =9$
3. The equation of a line can be written in the form $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept

STEP 2

First, we need to find the slope of the given line. We can do this by rearranging the equation into slope-intercept form ( $y = mx + c$ ), where $m$ is the slope.
$-x +4y =9$ $4y =x +9$ $y = \frac{}{4}x + \frac{9}{4}$ So, the slope of the given line is $\frac{}{4}$ .

STEP 3

The slope of the line perpendicular to the given line is the negative reciprocal of the slope of the given line. We can find this by flipping the fraction and changing the sign.
$m_{perpendicular} = -\frac{1}{m_{given}}$

STEP 4

Plug in the value for the slope of the given line to calculate the slope of the line perpendicular to it.
$m_{perpendicular} = -\frac{1}{\frac{3}{4}}$

STEP 5

Calculate the slope of the line perpendicular to the given line.
$m_{perpendicular} = -\frac{4}{3}$

STEP 6

Now that we have the slope of the line we are trying to find, we can use the point-slope form of a line to find the equation of the line. The point-slope form is $y - y1 = m(x - x1)$ , where $(x1, y1)$ is a point on the line and $m$ is the slope.
$y - y1 = m_{perpendicular}(x - x1)$

STEP 7

Plug in the values for the point and the slope to find the equation of the line.
$y -1 = -\frac{4}{3}(x - (-4))$

STEP 8

implify the equation.
$y -1 = -\frac{4}{3}x - \frac{16}{3}$

STEP 9

Rearrange the equation into slope-intercept form ( $y = mx + c$ ).
$y = -\frac{4}{3}x - \frac{16}{3} +$

STEP 10

implify the equation.
$y = -\frac{4}{3}x - \frac{13}{3}$ So, the equation of the line that passes through the point $(-4,)$ and that is perpendicular to the line with the equation $-3x +4y =9$ is $y = -\frac{4}{3}x - \frac{13}{3}$ .

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