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The equation for line $t$ can be written as $y = 4x - 8$ . Line $u$ , which is perpendicular to line $t$ , includes the point $(8,8)$ . What is the equation of line $u$ ? 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 for line

t

can be written as

y = 4x - 8

. Line

u

, which is perpendicular to line

t

, includes the point

(8,8)

. What is the equation of line

u

? 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 t: Determine the slope of line t. $\newline$ The equation of line t is given as $y = 4x - 8$ . The slope-intercept form of a line is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Comparing the given equation with the slope-intercept form, we find that the slope ( $m$ ) of line t is $4$ .
Find slope of line $u$ : Find the slope of line $u$ . Since line $u$ is perpendicular to line $t$ , its slope will be the negative reciprocal of the slope of line $t$ . The negative reciprocal of $4$ is $-\frac{1}{4}$ . Therefore, the slope of line $u$ is $-\frac{1}{4}$ .
Use point-slope form: Use the point-slope form to find the equation of line $u$ . We have the slope of line $u$ $-\frac{1}{4}$ and a point through which it passes $(8,8)$ . The point-slope form of a line is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point on the line. Plugging in the values, we get $y - 8 = -\frac{1}{4}(x - 8)$ .
Simplify to slope-intercept form: Simplify the equation to slope-intercept form. Starting with $y - 8 = -\frac{1}{4}(x - 8)$ , we distribute the slope on the right side to get $y - 8 = -\frac{1}{4}x + 2$ . Then, we add $8$ to both sides to solve for $y$ , resulting in $y = -\frac{1}{4}x + 2 + 8$ . Simplifying further, we get $y = -\frac{1}{4}x + 10$ .

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