In the xy-plane, a line has a y-intercept of (0, d) and passes through the point (4, d - 3… — SAT

In the $xy$ -plane, a line has a $y$ -intercept of $(0, d)$ and passes through the point $(4, d - 3)$ , where $d$ is a constant. If the line also passes through the point $(12, k)$ , which of the following expressions represents the value of $k$ ?

A
$d + 9$
B
$d - 16$
$d - 9$ Answer
D
$d - 36$

Answer

d - 9

The correct answer represents the value of

k

d - 9

. The slope

m

of the line passing through

(0, d)

and

(4, d - 3)

\frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

. Using the

y

-intercept

(0, d)

, the equation of the line is

y = -\frac{3}{4}x + d

. Substituting the point

(12, k)

gives

k = -\frac{3}{4}(12) + d = d - 9

Step-by-Step Solution

Calculate the slope of the line using the points

(0, d)

and

(4, d - 3)

m = -\frac{3}{4}

The slope formula is

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

. Substituting the given coordinates yields

m = \frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

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

y = -\frac{3}{4}x + d

Since the line's

y

-intercept is

(0, d)

, the constant

b

in the slope-intercept equation

y = mx + b

is equal to

d

Substitute the point

(12, k)

into the equation of the line to solve for

k

k = d - 9

Substituting

x = 12

and

y = k

into

y = -\frac{3}{4}x + d

gives

k = -\frac{3}{4}(12) + d = -9 + d

, which simplifies to

d - 9

Key Concept

Determining the equation and coordinates of a line in the coordinate plane given its slope and intercepts.

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