Question

which set of ordered pairs $(x,y)$ could represent a linear function?
$mathbf{a} = \\{(-6,-8), (-2,-5), (1,-2), (4,1)\\}$
$mathbf{b} = \\{(-2,1), (1,2), (4,3), (7,4)\\}$
$mathbf{c} = \\{(-2,3), (0,0), (2,-4), (4,-7)\\}$
$mathbf{d} = \\{(-5,1), (0,3), (4,5), (8,7)\\}$
answer
$\bigcirc$ a
$\bigcirc$ b
$\bigcirc$ c
$\bigcirc$ d

Explanation:

Step1: Recall linear function rule

A set of ordered pairs represents a linear function if the rate of change (slope) between every pair of points is constant. The slope formula is $m=\frac{y_2-y_1}{x_2-x_1}$.

Step2: Calculate slopes for Set A

Between $(-6,-8)$ and $(-2,-5)$: $m=\frac{-5-(-8)}{-2-(-6)}=\frac{3}{4}$
Between $(-2,-5)$ and $(1,-2)$: $m=\frac{-2-(-5)}{1-(-2)}=\frac{3}{3}=1$
Slopes are not equal, so not linear.

Step3: Calculate slopes for Set B

Between $(-2,1)$ and $(1,2)$: $m=\frac{2-1}{1-(-2)}=\frac{1}{3}$
Between $(1,2)$ and $(4,3)$: $m=\frac{3-2}{4-1}=\frac{1}{3}$
Between $(4,3)$ and $(7,4)$: $m=\frac{4-3}{7-4}=\frac{1}{3}$
Slopes are constant, check other sets to confirm.

Step4: Calculate slopes for Set C

Between $(-2,3)$ and $(0,0)$: $m=\frac{0-3}{0-(-2)}=-\frac{3}{2}$
Between $(0,0)$ and $(2,-4)$: $m=\frac{-4-0}{2-0}=-2$
Slopes are not equal, so not linear.

Step5: Calculate slopes for Set D

Between $(-5,1)$ and $(0,3)$: $m=\frac{3-1}{0-(-5)}=\frac{2}{5}$
Between $(0,3)$ and $(4,5)$: $m=\frac{5-3}{4-0}=\frac{2}{4}=\frac{1}{2}$
Slopes are not equal, so not linear.

Answer:

B. $\{(-2, 1), (1, 2), (4, 3), (7, 4)\}$