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

Cevap

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

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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