In the xy-plane, the graph of the linear function f passes through the points (k, 12) and … — SAT

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(k, 12)$ and $(8, k)$ , where $k$ is a constant. If the slope of the graph of $f$ is $-\frac{1}{3}$ , what is the value of $k$ ?

Answer: 14

Answer

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Substituting the given points

(k, 12)

and

(8, k)

and the slope

-\frac{1}{3}

into the formula gives

-\frac{1}{3} = \frac{k - 12}{8 - k}

. Cross-multiplying yields

3(k - 12) = -1(8 - k)

, which simplifies to

3k - 36 = -8 + k

. Isolating the variable

k

gives

2k = 28

, so

k = 14

Step-by-Step Solution

Apply the slope formula

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

using the points

(k, 12)

and

(8, k)

with slope

-\frac{1}{3}

-\frac{1}{3} = \frac{k - 12}{8 - k}

The slope of a linear function is constant and defined by the ratio of the change in

y

-values to the change in

x

-values.

Cross-multiply to solve the equation for

k

3(k - 12) = -1(8 - k)

To eliminate the fractions and solve the rational equation.

Distribute and simplify both sides of the equation.

3k - 36 = -8 + k

Distributing the constants on both sides prepares the equation for isolating the variable

k

Isolate the variable

k

by subtracting

k

from both sides and adding

36

to both sides.

2k = 28

Grouping like terms on opposite sides of the equation.

Divide by

2

to find the value of

k

k = 14

To find the final numerical value of the constant

k

Key Concept

Linear function slope formula

Estimated Time:1m 30s

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