In the xy-plane, the graph of a linear function f passes through the points (k, 2k + 3) an… — SAT

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(k, 2k + 3)$ and $(2k, 5k - 1)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $f$ is $-9$ , what is the value of $k$ ?

Answer: 16

Answer

The value of

k

is 16.

The value of

k

is 16 because when

k = 16

, the points on the graph are

(16, 35)

and

(32, 79)

. The slope of the line is

\frac{79 - 35}{32 - 16} = 2.75

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

y = 2.75x + b

. Using the point

(16, 35)

, we get

35 = 2.75(16) + b \implies 35 = 44 + b \implies b = -9

, which matches the given

y

-intercept of

-9

Step-by-Step Solution

Calculate the slope of the line in terms of

k

using the two given points

(k, 2k + 3)

and

(2k, 5k - 1)

The slope

m

\frac{3k - 4}{k}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

Use the

y

-intercept of

-9

, which corresponds to the point

(0, -9)

, along with the point

(k, 2k + 3)

to write another expression for the slope.

The slope

m

\frac{2k + 12}{k}

The slope must be constant for all points on the line, so the slope between the

y

-intercept and one of the points must equal the slope between the two points.

Equate the two slope expressions and solve for

k

k = 16

Setting the two expressions for the slope equal to each other gives

\frac{3k - 4}{k} = \frac{2k + 12}{k}

. Multiplying by

k

on both sides yields

3k - 4 = 2k + 12

, which simplifies to

k = 16

Key Concept

Linear Functions and Graphs

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