In the standard (x,y) coordinate plane, line q is perpendicular to the line with equation … — ACT

In the standard $(x,y)$ coordinate plane, line $q$ is perpendicular to the line with equation $y = \frac{3}{4}x - 2$ . If line $q$ passes through the point $(2, 1)$ , what is the $y$ -intercept of line $q$ ?

A
$-\frac{5}{3}$
B
$\frac{9}{4}$
$\frac{11}{3}$ Answer
D
$\frac{2}{3}$
E
$-\frac{1}{2}$

Answer

The

y

-intercept of line

q

\frac{11}{3}

The slope of the given line

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

\frac{3}{4}

. The slope of a perpendicular line is the negative reciprocal of the original slope, which is

-\frac{4}{3}

. Using the point-slope form of a linear equation with the point

(2, 1)

gives

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

. To find the

y

-intercept, set

x = 0

, which gives

y - 1 = -\frac{4}{3}(-2) = \frac{8}{3}

. Adding

1

to both sides yields

y = \frac{8}{3} + 1 = \frac{11}{3}

. Therefore, the correct

y

-intercept is

\frac{11}{3}

Step-by-Step Solution

Determine the slope of the perpendicular line

q

The slope of the given line is

\frac{3}{4}

, so the slope of line

q

(the negative reciprocal) is

-\frac{4}{3}

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

Write the equation of line

q

using the point-slope form with the point

(2, 1)

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

The point-slope form

y - y_1 = m(x - x_1)

is used to find the equation of a line given its slope and a point on the line.

Find the

y

-intercept by setting

x = 0

and solving for

y

y - 1 = -\frac{4}{3}(0 - 2) \Rightarrow y - 1 = \frac{8}{3} \Rightarrow y = \frac{8}{3} + 1 = \frac{11}{3}

The

y

-intercept of a line is the value of

y

when

x = 0

Key Concept

The slope of a line perpendicular to a line with slope

m

-\frac{1}{m}

. The equation of a line can be written in point-slope form as

y - y_1 = m(x - x_1)

Question

Answer

Step-by-Step Solution

Key Concept

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