Question

question
which set of ordered pairs ((x, y)) could represent a linear function?

( mathbf{a} = {(-6, 9), (-3, 6), (0, 3), (4, 0)} )

( mathbf{b} = {(1, 1), (2, 3), (4, 5), (5, 7)} )

( mathbf{c} = {(-4, -4), (-2, -2), (1, 0), (4, 3)} )

( mathbf{d} = {(3, 7), (5, 1), (6, -2), (7, -5)} )

answer
( circ ) a ( circ ) b
( circ ) c ( circ ) d
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Explanation:

Step1: Recall linear function slope rule

A linear function has a constant slope between all pairs of points. Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$

Step2: Calculate slopes for Set A

Between $(-6,9)$ & $(-3,6)$: $m=\frac{6-9}{-3-(-6)}=\frac{-3}{3}=-1$
Between $(-3,6)$ & $(0,3)$: $m=\frac{3-6}{0-(-3)}=\frac{-3}{3}=-1$
Between $(0,3)$ & $(4,0)$: $m=\frac{0-3}{4-0}=\frac{-3}{4}=-0.75$
Slopes are not constant.

Step3: Calculate slopes for Set B

Between $(1,1)$ & $(2,3)$: $m=\frac{3-1}{2-1}=2$
Between $(2,3)$ & $(4,5)$: $m=\frac{5-3}{4-2}=1$
Slopes are not constant.

Step4: Calculate slopes for Set C

Between $(-4,-4)$ & $(-2,-2)$: $m=\frac{-2-(-4)}{-2-(-4)}=1$
Between $(-2,-2)$ & $(1,0)$: $m=\frac{0-(-2)}{1-(-2)}=\frac{2}{3}$
Slopes are not constant.

Step5: Calculate slopes for Set D

Between $(3,7)$ & $(5,1)$: $m=\frac{1-7}{5-3}=\frac{-6}{2}=-3$
Between $(5,1)$ & $(6,-2)$: $m=\frac{-2-1}{6-5}=-3$
Between $(6,-2)$ & $(7,-5)$: $m=\frac{-5-(-2)}{7-6}=-3$
Slopes are constant.

Answer:

D. $\{(3, 7), (5, 1), (6, -2), (7, -5)\}$