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Line $t$ has an equation of $y = \frac{4}{3}x - 2$ . Line $u$ is perpendicular to line $t$ and passes through $(-2,2)$ . What is the equation of line $u$ ? $\newline$ 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. Line

t

has an equation of

y = \frac{4}{3}x - 2

. Line

u

is perpendicular to line

t

and passes through

(-2,2)

. What is the equation of line

u

?

\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 t: Determine the slope of line t. $\newline$ The equation of line t is given as $y = \frac{4}{3}x - 2$ . The slope-intercept form of a line is $y = mx + b$ , where $m$ is the slope. Therefore, the slope of line t is $\frac{4}{3}$ .
Find slope of line $u$ : Find the slope of line $u$ . Since line $u$ is perpendicular to line $t$ , the slope of line $u$ will be the negative reciprocal of the slope of line $t$ . The negative reciprocal of $\frac{4}{3}$ is $-\frac{3}{4}$ .
Use point-slope form: Use the point-slope form to find the equation of line $u$ . Line $u$ passes through the point $(-2,2)$ and has a slope of $-\frac{3}{4}$ . 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 a point on the line. Plugging in the values, we get $y - 2 = -\frac{3}{4}(x - (-2))$ .
Simplify equation of line u: Simplify the equation of line u. Expanding the equation from Step $3$ , we get $y - 2 = -\frac{3}{4}(x + 2)$ . Distributing the slope, we have $y - 2 = -\frac{3}{4}x - \frac{3}{4}(2)$ . Simplifying further, $y - 2 = -\frac{3}{4}x - \frac{3}{2}$ .
Solve for y: Solve for y to put the equation in slope-intercept form. $\newline$ Adding $2$ to both sides of the equation to solve for y, we get $y = -\frac{3}{4}x - \frac{3}{2} + 2$ . To combine the constant terms, we need a common denominator, which is $2$ . So, $2$ is the same as $\frac{4}{2}$ , and we have $y = -\frac{3}{4}x - \frac{3}{2} + \frac{4}{2}$ .
Combine constant terms: Combine the constant terms to find the y-intercept. Combining $-\frac{3}{2}$ and $\frac{4}{2}$ , we get $y = -\frac{3}{4}x + \frac{1}{2}$ . This is the equation of line $u$ in slope-intercept form.

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