In the xy-plane, line L is given by the equation 4x + 3y = 24. Line M passes through the o… — GRE General Test

In the $xy$ -plane, line $L$ is given by the equation $4x + 3y = 24$ . Line $M$ passes through the origin $(0,0)$ and is perpendicular to line $L$ . Which of the following statements must be true? Select all that apply.

Line $M$ passes through the point $(8, 6)$ .Answer
The point of intersection of line $L$ and line $M$ lies in Quadrant I.Answer
C
The slope of line $M$ is $-\frac{3}{4}$ .
D
The area of the triangular region bounded by line $L$ and the coordinate axes is $48$ square units.
E
The $x$ -intercept of line $L$ is $(0, 8)$ .

Answer

The statements confirming that line

M

passes through

(8, 6)

and that the intersection point of lines

L

and

M

lies in Quadrant I are correct.

The correct statements accurately calculate the negative reciprocal slope of line

M

\frac{3}{4}

, verify that

(8,6)

satisfies

y = \frac{3}{4}x

, and correctly find that the intersection of lines

L

and

M

has positive

x

and

y

coordinates in Quadrant I.

Step-by-Step Solution

Find the slope and equation of line

L

Line

L

in slope-intercept form is

y = -\frac{4}{3}x + 8

, with slope

m_L = -\frac{4}{3}

x

-intercept

(6,0)

, and

y

-intercept

(0,8)

Converting to slope-intercept form

y = mx + b

exposes the slope and intercepts of line

L

Determine the slope and equation of perpendicular line

M

Slope

m_M = -\frac{1}{m_L} = \frac{3}{4}

. Line

M

passes through the origin, so its equation is

y = \frac{3}{4}x

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

Evaluate the statement regarding point

(8,6)

on line

M

y = \frac{3}{4}(8) = 6

. The point

(8,6)

satisfies the equation of line

M

Substituting the coordinates into the line equation verifies point membership.

Find the intersection point of line

L

and line

M

Substituting

y = \frac{3}{4}x

into

4x + 3y = 24

yields

4x + 3\left(\frac{3}{4}x\right) = 24 \implies \frac{25}{4}x = 24 \implies x = \frac{96}{25}

. Then

y = \frac{72}{25}

Both

x

and

y

coordinates are positive, which confirms the intersection lies in Quadrant I.

Key Concept

Perpendicular Line Slopes and Intersections in Coordinate Geometry

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