QuestionFind the equation of line $r$ , which is perpendicular to $y=\frac{8}{3} x-1$ and passes through $(8,-7)$ .

Studdy Solution

STEP 1

Assumptions1. Line $q$ has the equation $y=\frac{8}{3}x-1$ . Line $r$ is perpendicular to line $q$
3. Line $r$ passes through the point $(8,-7)$ 4. The slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$

STEP 2

First, we need to find the slope of line $q$ . The slope is the coefficient of $x$ in the equation of the line.
In the equation $y=\frac{8}{}x-1$ , the slope of line $q$ is $\frac{8}{}$ .

STEP 3

Next, we need to find the slope of line $r$ . Since line $r$ is perpendicular to line $q$ , its slope is the negative reciprocal of the slope of line $q$ .
The slope of line $r$ is $-\frac{1}{m_q}$ , where $m_q$ is the slope of line $q$ .

STEP 4

Substitute the value of $m_q$ into the equation for the slope of line $r$ .
$m_r = -\frac{1}{\frac{8}{3}}$

STEP 5

Calculate the slope of line $r$ .
$m_r = -\frac{3}{8}$

STEP 6

Now that we have the slope of line $r$ , we can use the point-slope form of a line to write the equation of line $r$ . The point-slope form of a line is $y-y1=m(x-x1)$ , where $(x1,y1)$ is a point on the line and $m$ is the slope of the line.

STEP 7

Substitute the values of the slope and the point $(,-7)$ into the point-slope form of a line.
$y-(-7) = -\frac{3}{}(x-)$

STEP 8

implify the left side of the equation.
$y+7 = -\frac{3}{8}(x-8)$

STEP 9

istribute the $-\frac{3}{8}$ on the right side of the equation.
$y+7 = -\frac{3}{8}x +3$

STEP 10

Subtract $7$ from both sides of the equation to solve for $y$ .
$y = -\frac{3}{8}x +3 -7$

STEP 11

implify the right side of the equation.
$y = -\frac{3}{8}x -4$
The equation of line $r$ is $y = -\frac{3}{8}x -4$ .

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