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

In the standard $(x, y)$ coordinate plane, a line $L_1$ passes through the points $(2, 5)$ and $(-1, -4)$ . A second line, $L_2$ , passes through the $y$ -intercept of $L_1$ and has a slope that is twice the slope of $L_1$ . What is the $x$ -intercept of $L_2$ ?

$\frac{1}{6}$ Cevap
B
$-\frac{1}{6}$
C
$\frac{1}{3}$
D
$-\frac{3}{2}$
E
$-\frac{13}{2}$

Cevap

The

x

-intercept of

L_2

\frac{1}{6}

To find the

x

-intercept of

L_2

, the slope and

y

-intercept of

L_1

must first be found. The slope of

L_1

\frac{5 - (-4)}{2 - (-1)} = 3

. Using the point-slope form with the point

(2, 5)

, the equation of

L_1

y - 5 = 3(x - 2) \implies y = 3x - 1

, meaning its

y

-intercept is

(0, -1)

. The second line,

L_2

, has a slope of

2 \times 3 = 6

and passes through the same

y

-intercept

(0, -1)

, giving the equation

y = 6x - 1

. Setting

y = 0

to find the

x

-intercept yields

6x - 1 = 0 \implies x = \frac{1}{6}

Adım Adım Çözüm

Calculate the slope of line

L_1

using the coordinates

(2, 5)

and

(-1, -4)

The slope of

L_1

m_1 = \frac{5 - (-4)}{2 - (-1)} = \frac{9}{3} = 3

The slope of a line containing

(x_1, y_1)

and

(x_2, y_2)

is defined as

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

Determine the equation and the

y

-intercept of line

L_1

The equation of

L_1

in point-slope form is

y - 5 = 3(x - 2) \implies y = 3x - 1

. The

y

-intercept of

L_1

(0, -1)

Writing the equation in slope-intercept form

y = mx + b

directly identifies the

y

-intercept at

(0, b)

Determine the equation of line

L_2

The slope of

L_2

2 \times 3 = 6

. Since

L_2

passes through the

y

-intercept of

L_1

(0, -1)

, its equation is

y = 6x - 1

Using the relationship for the doubled slope and the shared

y

-intercept, the slope-intercept form of the second line is determined.

Find the

x

-intercept of

L_2

by setting

y = 0

Setting

y = 0

in the equation

y = 6x - 1

gives

0 = 6x - 1 \implies 6x = 1 \implies x = \frac{1}{6}

The

x

-intercept occurs where the line crosses the

x

-axis, which corresponds to setting

y = 0

Anahtar Kavram

Determining linear equations from coordinates, finding slopes, and calculating coordinate intercepts.

Soru

Cevap

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