Question

question
which set of ordered pairs $(x, y)$ could represent a linear function?
$\mathbf{a} = \\{(-5,2), (-3,1), (3,-2), (7,-3)\\}$
$\mathbf{b} = \\{(-9,2), (-3,4), (6,8), (9,9)\\}$
$\mathbf{c} = \\{(3,6), (5,2), (7,-3), (9,-7)\\}$
$\mathbf{d} = \\{(-4,-7), (0,-2), (4,3), (8,8)\\}$
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

First pair: $m_1=\frac{1-2}{-3-(-5)}=\frac{-1}{2}=-\frac{1}{2}$
Second pair: (missing y-value for (3, -z), cannot verify, so A is invalid)

Step3: Calculate slopes for Set B

First pair: $m_1=\frac{4-2}{-3-(-9)}=\frac{2}{6}=\frac{1}{3}$
Second pair: $m_2=\frac{8-4}{6-(-3)}=\frac{4}{9}$
$\frac{1}{3}
eq\frac{4}{9}$, so B is invalid.

Step4: Calculate slopes for Set C

First pair: $m_1=\frac{2-6}{5-3}=\frac{-4}{2}=-2$
Second pair: $m_2=\frac{-3-2}{7-5}=\frac{-5}{2}=-2.5$
$-2
eq-2.5$, so C is invalid.

Step5: Calculate slopes for Set D

First pair: $m_1=\frac{-2-(-7)}{0-(-4)}=\frac{5}{4}$
Second pair: $m_2=\frac{3-(-2)}{4-0}=\frac{5}{4}$
Third pair: $m_3=\frac{8-3}{8-4}=\frac{5}{4}$
All slopes are equal, so D is valid.

Answer:

D. $\{(-4,-7), (0,-2), (4,3), (8,8)\}$