QuestionFind the equation of line $t$ that is perpendicular to $y=-\frac{1}{9} x-4$ and passes through $(1,7)$ .

Studdy Solution

STEP 1

Assumptions1. The equation of line $s$ is $y=-\frac{1}{9} x-4$ . Line $t$ includes the point $(1,7)$ 3. Line $t$ is perpendicular to line $s$

STEP 2

The slope of line $s$ can be found from its equation. The equation of a line in slope-intercept form is $y=mx+b$ , where $m$ is the slope. In the equation of line $s$ , $y=-\frac{1}{9} x-4$ , the slope is $-\frac{1}{9}$ .

STEP 3

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the slope of line $t$ is the negative reciprocal of $-\frac{1}{9}$ .
$m_t = -\frac{1}{m_s}$

STEP 4

Substitute $-\frac{1}{9}$ for $m_s$ in the equation above to find the slope of line $t$ .
$m_t = -\frac{1}{-\frac{1}{9}}$

STEP 5

Calculate the slope of line $t$ .
$m_t = -\frac{1}{-\frac{1}{9}} =9$

STEP 6

The equation of a line in point-slope form is $y-y1=m(x-x1)$ , where $(x1,y1)$ is a point on the line and $m$ is the slope. We can use this form to write the equation of line $t$ since we know a point on the line $(1,)$ and the slope $9$ .

STEP 7

Substitute the values into the point-slope form of the equation to find the equation of line $t$ .
$y-7=9(x-1)$

STEP 8

Expand the right side of the equation.
$y-7=x-$

STEP 9

Rearrange the equation to put it in slope-intercept form $y=mx+b$ .
$y=9x-2$ The equation of line $t$ is $y=9x-2$ .

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