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 $k$ ?

Answer: 6

Answer

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

Step-by-Step Solution

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

Key Concept

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

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