QuestionFind the equation of line $r$ that is perpendicular to $y=3x-2$ and passes through the point $(1,3)$ .

Studdy Solution

STEP 1

Assumptions1. The equation of line $q$ is $y=3x-$ . Line $r$ is perpendicular to line $q$
3. Line $r$ passes through the point $(1,3)$ 4. The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line

STEP 2

First, we need to find the slope of line $q$ . The slope of a line in the form $y=mx+b$ is $m$ .
$lope_{q} =$

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 $q$ .
$lope_{r} = -1/Slope_{q}$

STEP 4

Now, plug in the value for the slope of line $q$ to calculate the slope of line $r$ .
$lope_{r} = -1/3$

STEP 5

The equation of a line in slope-intercept form is $y=mx+b$ , where $m$ is the slope and $b$ is the y-intercept. We know the slope of line $r$ and a point $(x1, y1)$ on the line, so we can use the point-slope form of a line to find the y-intercept. The point-slope form is $y-y1=m(x-x1)$ .
$3 - y1 = -1/3 * (1 - x1)$

STEP 6

Now, plug in the given point $(1,3)$ for $(x1, y1)$ to find the y-intercept.
$3 -3 = -1/3 * (1 -1)$

STEP 7

implify the equation to find the y-intercept.
$0 =0$

STEP 8

Now that we have the y-intercept, we can write the equation of line $r$ in slope-intercept form.
$y = Slope_{r} * x + y-intercept$

STEP 9

Plug in the values for the slope and the y-intercept to write the equation of line $r$ .
$y = -/3 * x +$

STEP 10

implify the equation to get the final equation of line $r$ .
$y = -/3 * x$ The equation of line $r$ is $y = -/3x$ .

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