Line k in the coordinate plane is represented by the equation 4x - 6y = 15. If Line p is p… — ACT

Line $k$ in the coordinate plane is represented by the equation $4x - 6y = 15$ . If Line $p$ is perpendicular to Line $k$ , what is the slope of Line $p$ ?

$-\frac{3}{2}$ Answer
B
$-\frac{2}{3}$
C
$\frac{2}{3}$
D
$\frac{3}{2}$
E
$-\frac{15}{4}$

Answer

The slope of Line

p

-\frac{3}{2}

To find the slope of a line perpendicular to

4x - 6y = 15

, first convert the equation to slope-intercept form (

y = mx + b

). Solving for

y

yields

y = \frac{2}{3}x - \frac{5}{2}

, indicating that the original line has a slope of

\frac{2}{3}

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

\frac{2}{3}

-\frac{3}{2}

, which is the correct answer.

Step-by-Step Solution

Convert the equation of Line

k

into slope-intercept form (

y = mx + b

Subtract

4x

from both sides to get

-6y = -4x + 15

. Divide all terms by

-6

to get

y = \frac{4}{6}x - \frac{15}{6}

, which simplifies to

y = \frac{2}{3}x - \frac{5}{2}

Converting to slope-intercept form directly reveals the slope

m

as the coefficient of

x

Identify the slope of Line

k

The slope of Line

k

m_k = \frac{2}{3}

The coefficient of

x

y = \frac{2}{3}x - \frac{5}{2}

represents the slope.

Calculate the negative reciprocal of the slope to find the slope of perpendicular Line

p

The slope of Line

p

m_p = -\frac{1}{m_k} = -\frac{1}{2/3} = -\frac{3}{2}

Perpendicular lines in the coordinate plane have slopes whose product is

-1

(negative reciprocals).

Key Concept

The slope of a line given by

Ax + By = C

m = -\frac{A}{B}

. Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals (

m_1 \cdot m_2 = -1

Estimated Time:1m 15s

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