Line L is defined by the equation 2x + y = 6 in the xy-plane. Which of the following state… — GRE General Test

Line $L$ is defined by the equation $2x + y = 6$ in the $xy$ -plane. Which of the following statements about line $L$ must be true? Select all that apply.

The slope of line $L$ is $-2$ .Answer
The $x$ -intercept of line $L$ is $(3, 0)$ .Answer
C
The slope of line $L$ is $-\frac{1}{2}$ .
D
A line perpendicular to line $L$ has a slope of $-2$ .
E
The $y$ -intercept of line $L$ is $(0, -6)$ .

Answer

The slope of line

L

-2

, and the

x

-intercept of line

L

(3, 0)

The given line equation

2x + y = 6

can be rewritten in slope-intercept form

y = -2x + 6

. This directly shows that the slope is

-2

. Setting

y = 0

gives

2x = 6 \implies x = 3

, so the

x

-intercept is

(3, 0)

. Therefore, both the statement that the slope is

-2

and the statement that the

x

-intercept is

(3, 0)

are correct.

Step-by-Step Solution

Convert the equation to slope-intercept form (

y = mx + b

y = -2x + 6

This isolates

y

to clearly reveal the slope

m = -2

and the

y

-intercept

(0, 6)

Find the

x

-intercept by setting

y = 0

2x + 0 = 6 \implies x = 3

, giving coordinate

(3, 0)

The

x

-intercept is the point where the line crosses the

x

-axis.

Determine perpendicular slope rules.

Perpendicular slope

= -\frac{1}{-2} = \frac{1}{2}

Perpendicular lines have negative reciprocal slopes, not identical slopes.

Key Concept

Linear equations, slope-intercept form, intercepts, and perpendicular slopes in coordinate geometry.

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