Question

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

Explanation:

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

Step 1: Check Set A

• Between \((-3, 5)\) and \((0, 2)\): \( m = \frac{2 - 5}{0 - (-3)} = \frac{-3}{3} = -1 \)
• Between \((0, 2)\) and \((3, -1)\): \( m = \frac{-1 - 2}{3 - 0} = \frac{-3}{3} = -1 \)
• Between \((3, -1)\) and \((7, -4)\): \( m = \frac{-4 - (-1)}{7 - 3} = \frac{-3}{4} = -0.75 \)

The slope is not constant (last slope is different), so A is not linear.

Step 2: Check Set B

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

The slope is not constant (last slope is different), so B is not linear.

Step 3: Check Set C

• Between \((-3, 8)\) and \((-1, 7)\): \( m = \frac{7 - 8}{-1 - (-3)} = \frac{-1}{2} = -0.5 \)
• Between \((-1, 7)\) and \((1, 6)\): \( m = \frac{6 - 7}{1 - (-1)} = \frac{-1}{2} = -0.5 \)
• Between \((1, 6)\) and \((3, 5)\): \( m = \frac{5 - 6}{3 - 1} = \frac{-1}{2} = -0.5 \)

The slope is constant (\(-0.5\)) for all consecutive points.

Step 4: Check Set D (for completeness)

• Between \((-5, -1)\) and \((-2, 2)\): \( m = \frac{2 - (-1)}{-2 - (-5)} = \frac{3}{3} = 1 \)
• Between \((-2, 2)\) and \((1, 4)\): \( m = \frac{4 - 2}{1 - (-2)} = \frac{2}{3} \approx 0.666 \)

The slope is not constant, so D is not linear.

Answer:

C. \(\{(-3, 8),\ (-1, 7),\ (1, 6),\ (3, 5)\}\)