Line P is perpendicular to a line with a slope of 3. If line P passes through the point (2… — ACT

Line $P$ is perpendicular to a line with a slope of $3$ . If line $P$ passes through the point $(2, 4)$ , what is the $y$ -intercept of line $P$ ?

The

y

-intercept of line

P

\frac{14}{3}

The slope of a line perpendicular to a line with a slope of

3

is its negative reciprocal,

-\frac{1}{3}

. Using the point-slope form with the point

(2, 4)

gives the equation

y - 4 = -\frac{1}{3}(x - 2)

. Setting

x = 0

to find the

y

-intercept yields

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

, which simplifies to

y = \frac{14}{3}

Determine the slope of line

P

The slope of line

P

-\frac{1}{3}

Perpendicular lines have slopes that are negative reciprocals. The negative reciprocal of

3

-\frac{1}{3}

Write the equation of line

P

in point-slope form using the point

(2, 4)

y - 4 = -\frac{1}{3}(x - 2)

The point-slope form is

y - y_1 = m(x - x_1)

, where

m

is the slope and

(x_1, y_1)

is the given point.

Solve for the

y

-intercept by setting

x = 0

y = \frac{14}{3}

The

y

-intercept occurs where

x = 0

. Substituting

x = 0

gives

y - 4 = -\frac{1}{3}(0 - 2) \Rightarrow y - 4 = \frac{2}{3} \Rightarrow y = \frac{2}{3} + 4 = \frac{14}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

Question