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The equation of line $t$ is $y = \frac{7}{4}x + 2$ . Perpendicular to line $t$ is line $u$ , which passes through the point $(2,-2)$ . 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 of line

t

is

y = \frac{7}{4}x + 2

. Perpendicular to line

t

is line

u

, which passes through the point

(2,-2)

. 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 $y = \frac{7}{4}x + 2$ . The slope ( $m$ ) of a line in the slope-intercept form $y = mx + b$ is the coefficient of $x$ . Therefore, the slope of line t is $\frac{7}{4}$ .
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 $\frac{7}{4}$ is $-\frac{4}{7}$ . Therefore, the slope of line u is $-\frac{4}{7}$ .
Use point-slope form: Use the point-slope form to find the equation of line $u$ . We have the slope of line $u$ $-\frac{4}{7}$ and a point through which it passes $(2,-2)$ . The point-slope form of a line's equation 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 $y - (-2) = -\frac{4}{7}(x - 2)$ .
Simplify to slope-intercept form: Simplify the equation to slope-intercept form. $\newline$ Starting with $y + 2 = -\frac{4}{7}(x - 2)$ , distribute the slope $-\frac{4}{7}$ across $(x - 2)$ to get $y + 2 = -\frac{4}{7}x + \frac{8}{7}$ . Then, subtract $2$ from both sides to isolate $y$ , which gives us $y = -\frac{4}{7}x + \frac{8}{7} - \frac{14}{7}$ . Simplifying the y-intercept, we get $y = -\frac{4}{7}x - \frac{6}{7}$ .

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