hich set of ordered pairs left parenthesis, x, comma, y, right parenthesis(x,y) could represent a linear function?

Question

A, equals, left curly bracket, left parenthesis, minus, 3, comma, 8, right parenthesis, comma, left parenthesis, 0, comma, 7, right parenthesis, comma, left parenthesis, 4, comma, 6, right parenthesis, comma, left parenthesis, 8, comma, 5, right parenthesis, right curly bracket
A=
{(−3,8),(0,7),(4,6),(8,5)}
B, equals, left curly bracket, left parenthesis, 1, comma, 7, right parenthesis, comma, left parenthesis, 2, comma, 4, right parenthesis, comma, left parenthesis, 3, comma, 1, right parenthesis, comma, left parenthesis, 6, comma, minus, 8, right parenthesis, right curly bracket
B=
{(1,7),(2,4),(3,1),(6,−8)}
C, equals, left curly bracket, left parenthesis, minus, 2, comma, minus, 2, right parenthesis, comma, left parenthesis, 1, comma, minus, 1, right parenthesis, comma, left parenthesis, 4, comma, 1, right parenthesis, comma, left parenthesis, 7, comma, 3, right parenthesis, right curly bracket
C=
{(−2,−2),(1,−1),(4,1),(7,3)}
D, equals, left curly bracket, left parenthesis, minus, 5, comma, 7, right parenthesis, comma, left parenthesis, minus, 2, comma, 6, right parenthesis, comma, left parenthesis, 1, comma, 5, right parenthesis, comma, left parenthesis, 3, comma, 4, right parenthesis, right curly bracket
D=
{(−5,7),(−2,6),(1,5),(3,4)}
Answer
Multiple Choice Answers
AA
BB
CC
DD

Answer 1

To determine which set of ordered pairs represents a linear function, we need to check if the pairs can be connected by a straight line. This can be verified by checking if the slope between any two points remains consistent.

Let's analyze each set:

Set A: \[ A = {(-3, 8), (0, 7), (4, 6), (8, 5)} \] Calculating the slopes:

• Between \((-3, 8)\) and \((0, 7)\): \[ \text{slope} = \frac{7 - 8}{0 - (-3)} = \frac{-1}{3} \]
• Between \((0, 7)\) and \((4, 6)\): \[ \text{slope} = \frac{6 - 7}{4 - 0} = \frac{-1}{4} \]
• Between \((4, 6)\) and \((8, 5)\): \[ \text{slope} = \frac{5 - 6}{8 - 4} = \frac{-1}{4} \]

The slopes are not consistent. Thus, set A does not represent a linear function.

Set B: \[ B = {(1, 7), (2, 4), (3, 1), (6, -8)} \] Calculating the slopes:

• Between \((1, 7)\) and \((2, 4)\): \[ \text{slope} = \frac{4 - 7}{2 - 1} = \frac{-3}{1} = -3 \]
• Between \((2, 4)\) and \((3, 1)\): \[ \text{slope} = \frac{1 - 4}{3 - 2} = \frac{-3}{1} = -3 \]
• Between \((3, 1)\) and \((6, -8)\): \[ \text{slope} = \frac{-8 - 1}{6 - 3} = \frac{-9}{3} = -3 \]

All slopes are consistent (-3). Therefore, set B represents a linear function.

Set C: \[ C = {(-2, -2), (1, -1), (4, 1), (7, 3)} \] Calculating the slopes:

• Between \((-2, -2)\) and \((1, -1)\): \[ \text{slope} = \frac{-1 - (-2)}{1 - (-2)} = \frac{1}{3} \]
• Between \((1, -1)\) and \((4, 1)\): \[ \text{slope} = \frac{1 - (-1)}{4 - 1} = \frac{2}{3} \]
• Between \((4, 1)\) and \((7, 3)\): \[ \text{slope} = \frac{3 - 1}{7 - 4} = \frac{2}{3} \]

The slopes are not consistent. Thus, set C does not represent a linear function.

Set D: \[ D = {(-5, 7), (-2, 6), (1, 5), (3, 4)} \] Calculating the slopes:

• Between \((-5, 7)\) and \((-2, 6)\): \[ \text{slope} = \frac{6 - 7}{-2 - (-5)} = \frac{-1}{3} \]
• Between \((-2, 6)\) and \((1, 5)\): \[ \text{slope} = \frac{5 - 6}{1 - (-2)} = \frac{-1}{3} \]
• Between \((1, 5)\) and \((3, 4)\): \[ \text{slope} = \frac{4 - 5}{3 - 1} = \frac{-1}{2} \]

The slopes are not consistent. Thus, set D does not represent a linear function.

Based on the analysis, the only set that represents a linear function is Set B.

So the answer is B.