In the xy-plane, line p is defined by the equation y = -\frac{3}{4}x + 3. Which of the fol… — GRE General Test

In the $xy$ -plane, line $p$ is defined by the equation $y = -\frac{3}{4}x + 3$ . Which of the following statements about line $p$ must be true? Select all that apply.

Line $p$ passes through the point $(4, 0)$ .Answer
Line $p$ is perpendicular to any line with a slope of $\frac{4}{3}$ .Answer
C
The $y$ -intercept of line $p$ is $(0, -3)$ .
D
Line $p$ passes through Quadrant III.
E
The slope of line $p$ is $-\frac{4}{3}$ .

Answer

The statements asserting that line

p

passes through the point

(4, 0)

and that line

p

is perpendicular to any line with a slope of

\frac{4}{3}

are correct.

Substituting

x = 4

into the line equation gives

y = 0

, showing that

(4, 0)

lies on line

p

. Additionally, the slope of line

p

-\frac{3}{4}

, and the negative reciprocal of

-\frac{3}{4}

\frac{4}{3}

, making any line with slope

\frac{4}{3}

perpendicular to line

p

Step-by-Step Solution

Identify the slope and

y

-intercept directly from the equation

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

The slope is

m = -\frac{3}{4}

and the

y

-intercept is

(0, 3)

The equation is given in standard slope-intercept form

y = mx + b

Verify point inclusion and perpendicular slope relationship.

Substituting

x = 4

yields

y = -\frac{3}{4}(4) + 3 = 0

, confirming

(4, 0)

is on the line. The negative reciprocal of

-\frac{3}{4}

\frac{4}{3}

, confirming the perpendicular line slope.

A point lies on a line if its coordinates satisfy the equation, and perpendicular lines have slopes that multiply to

-1

Analyze quadrant coverage.

Line

p

connects

(0, 3)

on the positive

y

-axis to

(4, 0)

on the positive

x

-axis, covering Quadrant I. For

x < 0

y > 3

(Quadrant II). For

x > 4

y < 0

(Quadrant IV). It never enters Quadrant III where both coordinates are negative.

A line with a positive

y

-intercept and negative slope crosses Quadrants I, II, and IV only.

Key Concept

Line properties in coordinate geometry including slope, intercepts, perpendicularity, and quadrant passage.

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