In the xy-plane, line l passes through the point (3, 4) and has a slope of m, where m > 0.… — SAT

In the $xy$ -plane, line $l$ passes through the point $(3, 4)$ and has a slope of $m$ , where $m > 0$ . Line $k$ is perpendicular to line $l$ and passes through the point $(2, 2)$ . If the sum of the $y$ -intercepts of line $l$ and line $k$ is $1$ , what is the value of $m$ ?

A
$\frac{1}{3}$
B
$\frac{2}{3}$
C
$1$
$2$ Cevap

Cevap

The correct value of

m

2

The correct answer is the value

2

. By finding the equations of both lines in slope-intercept form, we express their

y

-intercepts in terms of

m

: the

y

-intercept of line

l

4 - 3m

and the

y

-intercept of line

k

\frac{2}{m} + 2

. Setting their sum to

1

gives the equation

3m^2 - 5m - 2 = 0

. Factoring this quadratic yields the solutions

m = -\frac{1}{3}

and

m = 2

. Since

m

must be positive,

m = 2

is the only valid solution.

Adım Adım Çözüm

Find the equation and

y

-intercept of line

l

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 4)

and slope

m

, the equation of line

l

y - 4 = m(x - 3)

, which simplifies to

y = mx + 4 - 3m

. Thus, the

y

-intercept of line

l

4 - 3m

Expressing the

y

-intercept of line

l

in terms of

m

allows us to use it in the sum equation.

Find the equation and

y

-intercept of line

k

Since line

k

is perpendicular to line

l

, its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Using the point-slope form with the point

(2, 2)

, the equation of line

k

y - 2 = -\frac{1}{m}(x - 2)

, which simplifies to

y = -\frac{1}{m}x + \frac{2}{m} + 2

. Thus, the

y

-intercept of line

k

\frac{2}{m} + 2

Expressing the

y

-intercept of line

k

in terms of

m

allows us to use it in the sum equation.

Set up the equation for the sum of the

y

-intercepts and solve for

m

The sum of the

y

-intercepts is

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

. Simplifying this equation gives

6 - 3m + \frac{2}{m} = 1 \implies 5 - 3m + \frac{2}{m} = 0

. Multiplying by

m

yields

3m^2 - 5m - 2 = 0

. Factoring the quadratic gives

(3m + 1)(m - 2) = 0

, which has solutions

m = -\frac{1}{3}

and

m = 2

. Since

m > 0

, we have

m = 2

To determine the unique positive slope that satisfies the given conditions.

Anahtar Kavram

Writing equations of perpendicular lines and finding their intercepts using slopes and points.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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