In the xy-plane, line k is defined by the equation 2x + 4y = 9. Line m is perpendicular to… — GRE General Test

In the $xy$ -plane, line $k$ is defined by the equation $2x + 4y = 9$ . Line $m$ is perpendicular to line $k$ . What is the slope of line $m$ ?

2

First, rewrite the line equation

2x + 4y = 9

in slope-intercept form

y = mx + b

: subtracting

2x

from both sides gives

4y = -2x + 9

, and dividing by

4

yields

y = -\frac{1}{2}x + \frac{9}{4}

. Thus, the slope of line

k

m_k = -\frac{1}{2}

. Since line

m

is perpendicular to line

k

, its slope is the negative reciprocal of

-\frac{1}{2}

, which is

2

Convert the equation of line

k

to slope-intercept form (

y = mx + b

4y = -2x + 9 \implies y = -\frac{1}{2}x + \frac{9}{4}

The coefficient of

x

in slope-intercept form gives the slope of line

k

(

m_k = -\frac{1}{2}

Calculate the slope of perpendicular line

m

m_m = -\frac{1}{m_k} = -\frac{1}{-\frac{1}{2}} = 2

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

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

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