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The equation of line $s$ is $y = -\frac{1}{9}x - 4$ . Line $t$ includes the point $(1,7)$ and is perpendicular to line $s$ . What is the equation of line $t$ ? $\newline$ Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

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Write an equation for a parallel or perpendicular line

Full solution

Q. The equation of line

s

is

y = -\frac{1}{9}x - 4

. Line

t

includes the point

(1,7)

and is perpendicular to line

s

. What is the equation of line

t

?

\newline

Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Determine slope of line $s$ : Determine the slope of line $s$ . The equation of line $s$ is given as $y = -\frac{1}{9}x - 4$ . The slope ( $m$ ) of a line in the slope-intercept form $y = mx + b$ is the coefficient of $x$ . Therefore, the slope of line $s$ is $-\frac{1}{9}$ .
Find slope of line $t$ : Find the slope of line $t$ . Since line $t$ is perpendicular to line $s$ , its slope will be the negative reciprocal of the slope of line $s$ . The negative reciprocal of $-\frac{1}{9}$ is $9$ (because $-1/(-\frac{1}{9}) = 9$ ). Therefore, the slope of line $t$ is $9$ .
Use point-slope form: Use the point-slope form to find the equation of line $t$ . We have the slope of line $t$ ( $m = 9$ ) and a point on line $t$ ( $(1,7)$ ). The point-slope form of a line is $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line. Plugging in the values, we get $y - 7 = 9(x - 1)$ .
Simplify equation to slope-intercept form: Simplify the equation to slope-intercept form. $\newline$ To get the slope-intercept form $y = mx + b$ , we distribute the slope on the right side and add $7$ to both sides: $y - 7 = 9x - 9$ . Adding $7$ to both sides gives us $y = 9x - 9 + 7$ , which simplifies to $y = 9x - 2$ .

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