The graph of a linear equation in the xy-plane has an x-intercept of (k, 0) and a y-interc… — SAT

The graph of a linear equation in the $xy$ -plane has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, 3k)$ , where $k$ is a positive constant. If the line passes through the point $(2, 12)$ , what is the value of $k$ ?

Cevap: 6

Cevap

The value of

k

6

The correct answer is

6

. The slope of the line can be found using the two intercepts

(k, 0)

and

(0, 3k)

m = \frac{3k - 0}{0 - k} = -3

. The equation of the line in slope-intercept form is

y = -3x + 3k

. Since the line passes through the point

(2, 12)

, we substitute

x = 2

and

y = 12

into the equation to get

12 = -3(2) + 3k

. Simplifying the equation yields

12 = -6 + 3k

, which gives

18 = 3k

. Dividing both sides by

3

results in

k = 6

Adım Adım Çözüm

Find the slope of the line using the

x

-intercept

(k, 0)

and the

y

-intercept

(0, 3k)

The slope of the line is

-3

The slope formula is

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

. Substituting the points

(k, 0)

and

(0, 3k)

gives

m = \frac{3k - 0}{0 - k} = \frac{3k}{-k} = -3

Write the equation of the line in slope-intercept form.

The equation of the line is

y = -3x + 3k

Using the slope-intercept form

y = mx + b

, we substitute the slope

m = -3

and the

y

-intercept value

b = 3k

from the point

(0, 3k)

Substitute the given point

(2, 12)

into the equation and solve for

k

The value of

k

6

Substituting

x = 2

and

y = 12

into

y = -3x + 3k

yields

12 = -3(2) + 3k

. This simplifies to

12 = -6 + 3k

. Adding

6

to both sides gives

18 = 3k

, and dividing by

3

gives

k = 6

Anahtar Kavram

Linear equations in two variables, finding equations from intercepts, and constant determination via point substitution.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru