In the xy-plane, the graph of the linear function f passes through the points (-2, c) and … — SAT

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(-2, c)$ and $(c, 7)$ , where $c$ is a constant. If the slope of the graph of $f$ is $\frac{1}{2}$ , what is the value of $c$ ?

A
1
4Answer
C
6
D
16

Answer

The value of

c

is 4.

Using the slope formula

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

with the points

(-2, c)

and

(c, 7)

and the slope

m = \frac{1}{2}

gives

\frac{7 - c}{c - (-2)} = \frac{1}{2}

. Simplifying the denominator yields

\frac{7 - c}{c + 2} = \frac{1}{2}

. Cross-multiplying gives

2(7 - c) = c + 2

, which expands to

14 - 2c = c + 2

. Adding

2c

to both sides and subtracting

2

from both sides gives

12 = 3c

. Dividing both sides by

3

gives

c = 4

Step-by-Step Solution

Set up the slope formula for the line passing through

(-2, c)

and

(c, 7)

m = \frac{7 - c}{c - (-2)}

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}

Substitute the given slope

m = \frac{1}{2}

into the formula and simplify the denominator.

\frac{7 - c}{c + 2} = \frac{1}{2}

We are given that the slope of the line is

\frac{1}{2}

Solve the equation for

c

by cross-multiplying and isolating the variable.

2(7 - c) = c + 2 \implies 14 - 2c = c + 2 \implies 12 = 3c \implies c = 4

Cross-multiplying eliminates the fractions, allowing us to solve the linear equation for

c

Key Concept

Finding the slope of a line given two points containing an unknown coordinate

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