In the xy-plane, line L has a slope of -\frac{3}{4} and passes through the point (4, 1). W… — GRE General Test

In the $xy$ -plane, line $L$ has a slope of $-\frac{3}{4}$ and passes through the point $(4, 1)$ . Which of the following statements about line $L$ must be true? Select all such statements.

Line $L$ passes through the point $(-4, 7)$ .Answer
B
Line $L$ passes through Quadrant III.
Line $L$ is perpendicular to the line defined by $4x - 3y = 12$ .Answer
D
The $x$ -intercept of line $L$ is $\left(\frac{4}{3}, 0\right)$ .
Line $L$ is parallel to the line defined by $3x + 4y = 25$ .Answer

Answer

The true statements are that line L passes through (-4, 7), is perpendicular to 4x - 3y = 12, and is parallel to 3x + 4y = 25.

The correct statements are those identifying that

(-4, 7)

satisfies

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

, that

4x - 3y = 12

has a perpendicular slope of

\frac{4}{3}

, and that

3x + 4y = 25

has an identical parallel slope of

-\frac{3}{4}

Step-by-Step Solution

Find the equation of line L using point-slope form.

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

, or in standard form

3x + 4y = 16

Establishing the linear equation allows systematic verification of points, intercepts, and slope properties.

Verify point

(-4, 7)

on line L.

3(-4) + 4(7) = -12 + 28 = 16

, which satisfies the equation.

Checking if the point coordinates satisfy

3x + 4y = 16

confirms it lies on the line.

Determine quadrant coverage.

The

y

-intercept is

(0, 4)

and the

x

-intercept is

\left(\frac{16}{3}, 0\right)

. For

x < 0

y > 4

(Quadrant II); for

0 < x < \frac{16}{3}

y > 0

(Quadrant I); for

x > \frac{16}{3}

y < 0

(Quadrant IV).

Line L passes through Quadrants I, II, and IV only, so it does not enter Quadrant III.

Analyze slope relationships for perpendicularity and parallelism.

Line

4x - 3y = 12

has slope

\frac{4}{3}

(negative reciprocal of

-\frac{3}{4}

). Line

3x + 4y = 25

has slope

-\frac{3}{4}

(identical slope).

Perpendicular lines have slopes that multiply to

-1

, while parallel lines have equal slopes and different intercepts.

Key Concept

Linear line equations, parallel/perpendicular slopes, and point/intercept evaluations in coordinate geometry.

Estimated Time:1m 30s

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