Question

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

Explanation:

To determine which set of ordered pairs represents a linear function, we check if the slope between consecutive points is constant. The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 1: Check Set A

For \( A = \{(-6, 9), (-3, 7), (0, 5), (6, 1)\} \):

• Slope between \((-6, 9)\) and \((-3, 7)\): \( \frac{7 - 9}{-3 - (-6)} = \frac{-2}{3} = -\frac{2}{3} \)
• Slope between \((-3, 7)\) and \((0, 5)\): \( \frac{5 - 7}{0 - (-3)} = \frac{-2}{3} = -\frac{2}{3} \)
• Slope between \((0, 5)\) and \((6, 1)\): \( \frac{1 - 5}{6 - 0} = \frac{-4}{6} = -\frac{2}{3} \)

The slope is constant (\( -\frac{2}{3} \)) for all consecutive points in set A.

Step 2: Verify Other Sets (Optional, but for completeness)

• Set B: Slope between \((-5, 3)\) and \((-2, 5)\): \( \frac{5 - 3}{-2 - (-5)} = \frac{2}{3} \); Slope between \((-2, 5)\) and \((2, 7)\): \( \frac{7 - 5}{2 - (-2)} = \frac{2}{4} = \frac{1}{2} \). Not constant.
• Set C: Slope between \((-2, 3)\) and \((1, 0)\): \( \frac{0 - 3}{1 - (-2)} = -1 \); Slope between \((1, 0)\) and \((3, -1)\): \( \frac{-1 - 0}{3 - 1} = -\frac{1}{2} \). Not constant.
• Set D: Slope between \((-4, -1)\) and \((-1, 2)\): \( \frac{2 - (-1)}{-1 - (-4)} = 1 \); Slope between \((-1, 2)\) and \((2, 5)\): \( \frac{5 - 2}{2 - (-1)} = 1 \); Slope between \((2, 5)\) and \((5, 7)\): \( \frac{7 - 5}{5 - 2} = \frac{2}{3} \). Not constant (wait, earlier miscalculation? Wait, no—wait, D: (-4,-1), (-1,2): slope \( \frac{2 - (-1)}{-1 - (-4)} = \frac{3}{3} = 1 \); (-1,2) to (2,5): \( \frac{5 - 2}{2 - (-1)} = 1 \); (2,5) to (5,7): \( \frac{7 - 5}{5 - 2} = \frac{2}{3} \). So not constant. But wait, earlier Set A had constant slope. Wait, maybe I made a mistake with D. Wait, no—Set A: let's recheck. (-6,9) to (-3,7): \( (7-9)/(-3+6) = -2/3 \); (-3,7) to (0,5): (5-7)/(0+3) = -2/3; (0,5) to (6,1): (1-5)/(6-0) = -4/6 = -2/3. Yes, constant. So Set A has constant slope.

Answer:

A. \(\{(-6, 9), (-3, 7), (0, 5), (6, 1)\}\)