In the xy-plane, line m has an x-intercept of -6 and a y-intercept of 3. Line k is perpend… — GRE General Test

In the $xy$ -plane, line $m$ has an $x$ -intercept of $-6$ and a $y$ -intercept of $3$ . Line $k$ is perpendicular to line $m$ and passes through the point $(4, 9)$ . What is the $x$ -intercept of line $k$ ?

Answer: 8.5

Answer

8.5

First, determine the slope of line

m

using the given intercepts

(-6,0)

and

(0,3)

, yielding

\frac{3-0}{0-(-6)} = \frac{1}{2}

. Since line

k

is perpendicular to line

m

, its slope is the negative reciprocal of

\frac{1}{2}

, which is

-2

. Using the point-slope equation with point

(4,9)

, the equation of line

k

y - 9 = -2(x - 4)

, simplifying to

y = -2x + 17

. Finding the

x

-intercept by setting

y = 0

yields

0 = -2x + 17

, giving

x = 8.5

Step-by-Step Solution

Calculate the slope of line

m

Slope of line

m

\frac{1}{2}

Line

m

passes through the points

(-6,0)

and

(0,3)

Determine the slope of perpendicular line

k

Slope of line

k

-2

Perpendicular lines have negative reciprocal slopes.

Derive the equation of line

k

y = -2x + 17

Use point-slope form with given point

(4,9)

and slope

-2

Solve for the

x

-intercept of line

k

x = 8.5

Set

y = 0

in the linear equation

y = -2x + 17

Key Concept

Perpendicular lines, slope calculation from intercepts, and line equations

Question

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