In the xy-plane, line \ell_1 passes through the point (3, 7) and is perpendicular to line … — GRE General Test

In the $xy$ -plane, line $\ell_1$ passes through the point $(3, 7)$ and is perpendicular to line $\ell_2$ , which is defined by the equation $2x - 5y = 10$ . Line $\ell_3$ is parallel to line $\ell_1$ and has a $y$ -intercept that is $6$ units greater than the $y$ -intercept of line $\ell_1$ . If line $\ell_3$ intersects the $x$ -axis at the point $(a, 0)$ , what is the value of $a$ ?

Cevap: 8.2

Cevap

The value of

a

is 8.2 (or

\frac{41}{5}

Converting

2x - 5y = 10

to slope-intercept form yields

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

, so the slope of line

\ell_2

\frac{2}{5}

. Line

\ell_1

is perpendicular to

\ell_2

, giving it a slope of

m_1 = -\frac{5}{2}

. Using the point

(3, 7)

, the line equation for

\ell_1

y - 7 = -2.5(x - 3)

, which simplifies to

y = -2.5x + 14.5

. Line

\ell_3

is parallel to

\ell_1

, so

m_3 = -2.5

, and its

y

-intercept is

14.5 + 6 = 20.5

. Writing the equation for line

\ell_3

y = -2.5x + 20.5

and setting

y = 0

gives

0 = -2.5a + 20.5 \implies 2.5a = 20.5 \implies a = 8.2

Adım Adım Çözüm

Determine the slope of line

\ell_2

The slope of line

\ell_2

\frac{2}{5}

Convert

2x - 5y = 10

into slope-intercept form

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

Determine the slope of line

\ell_1

The slope of line

\ell_1

-\frac{5}{2}

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

Find the

y

-intercept of line

\ell_1

The

y

-intercept of

\ell_1

\frac{29}{2} = 14.5

Apply point-slope form with point

(3, 7)

y - 7 = -\frac{5}{2}(x - 3) \implies y = -\frac{5}{2}x + \frac{29}{2}

Construct the equation for line

\ell_3

The equation of

\ell_3

y = -\frac{5}{2}x + \frac{41}{2}

Line

\ell_3

has the same slope as

\ell_1

(

-\frac{5}{2}

) and its

y

-intercept is

14.5 + 6 = 20.5 = \frac{41}{2}

Calculate the

x

-intercept coordinate

a

of line

\ell_3

a = 8.2

Substitute

y = 0

into the equation for

\ell_3

0 = -\frac{5}{2}a + \frac{41}{2} \implies 5a = 41 \implies a = 8.2

Anahtar Kavram

Perpendicular and parallel slopes, point-slope equation derivation, line transformations, and intercept determination.

Tahmini Süre:2m 30s

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