In the standard (x, y) coordinate plane, line L_1 passes through the points (2, 5) and (6,… — ACT

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(2, 5)$ and $(6, -3)$ . A second line, $L_2$ , is perpendicular to $L_1$ and intersects $L_1$ at its $y$ -intercept. What is the $x$ -coordinate of the $x$ -intercept of $L_2$ ?

Answer: -18

Answer

The

x

-coordinate of the

x

-intercept of

L_2

-18

First, the slope of

L_1

is calculated as

-2

using the slope formula. Substituting one of the points into the slope-intercept form gives the

y

-intercept of

L_1

(0, 9)

. Since

L_2

is perpendicular to

L_1

, its slope is the negative reciprocal of

-2

, which is

\frac{1}{2}

. Since

L_2

shares the

y

-intercept

(0, 9)

, its equation is

y = \frac{1}{2}x + 9

. Setting

y = 0

to find the

x

-intercept gives

x = -18

Step-by-Step Solution

Calculate the slope of line

L_1

The slope of

L_1

-2

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the given points

(2, 5)

and

(6, -3)

yields

m_1 = \frac{-3 - 5}{6 - 2} = -2

Find the

y

-intercept of

L_1

The

y

-intercept is

(0, 9)

Substituting

m = -2

and the coordinates of

(2, 5)

into the slope-intercept equation

y = mx + b

gives

5 = -2(2) + b

, which simplifies to

b = 9

Find the slope of the perpendicular line,

L_2

The slope of

L_2

\frac{1}{2}

Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of

-2

\frac{1}{2}

Determine the equation of

L_2

and calculate its

x

-intercept.

The

x

-coordinate of the

x

-intercept of

L_2

-18

Since

L_2

passes through the

y

-intercept

(0, 9)

, its equation is

y = \frac{1}{2}x + 9

. Setting

y = 0

to find the

x

-intercept yields

0 = \frac{1}{2}x + 9

, which solves to

x = -18

Key Concept

Determining the equation and intercepts of a line perpendicular to a given line that passes through a specific shared point.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic