In the xy-plane, line k has a slope of \frac{2}{5} and passes through the point (5, 8). Li… — GRE General Test

In the $xy$ -plane, line $k$ has a slope of $\frac{2}{5}$ and passes through the point $(5, 8)$ . Line $m$ is perpendicular to line $k$ and has the same $y$ -intercept as line $k$ . What is the $x$ -intercept of line $m$ ?

Cevap: 2.4

Cevap

2.4

To find the

x

-intercept of line

m

, first determine the equation of line

k

. Using slope-intercept form

y = mx + b

with

m = \frac{2}{5}

and point

(5, 8)

, we get

8 = \frac{2}{5}(5) + b \implies b = 6

. Thus, the

y

-intercept of line

k

(and line

m

) is

(0, 6)

. Next, line

m

is perpendicular to line

k

, so its slope is the negative reciprocal of

\frac{2}{5}

, which is

-\frac{5}{2}

. The equation for line

m

y = -\frac{5}{2}x + 6

. Setting

y = 0

to solve for the

x

-intercept yields

0 = -\frac{5}{2}x + 6 \implies \frac{5}{2}x = 6 \implies x = \frac{12}{5} = 2.4

Adım Adım Çözüm

Find the equation and

y

-intercept of line

k

Line

k

has equation

y = \frac{2}{5}x + 6

, with

y

-intercept at

(0, 6)

Using point-slope form

y - y_1 = m(x - x_1)

with

m = \frac{2}{5}

and

(x_1, y_1) = (5, 8)

Calculate the slope of line

m

The slope of line

m

-\frac{5}{2}

Perpendicular lines have negative reciprocal slopes:

-\frac{1}{2/5} = -\frac{5}{2}

Construct the equation of line

m

Line

m

has equation

y = -\frac{5}{2}x + 6

Line

m

shares the

y

-intercept

(0, 6)

with line

k

Solve for the

x

-intercept of line

m

x = 2.4

Set

y = 0

y = -\frac{5}{2}x + 6

to get

0 = -\frac{5}{2}x + 6 \implies \frac{5}{2}x = 6 \implies x = 2.4

Anahtar Kavram

Perpendicular Slopes and Line Intercepts

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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