Question

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

Explanation:

Step1: Recall the property of linear functions

For a set of ordered pairs \((x,y)\) to represent a linear function, the slope between any two points should be constant. The slope formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Check Option A

• Between \((-1,7)\) and \((1,5)\): \(m_1=\frac{5 - 7}{1 - (-1)}=\frac{-2}{2}=- 1\)
• Between \((1,5)\) and \((3,3)\): \(m_2=\frac{3 - 5}{3 - 1}=\frac{-2}{2}=-1\)
• Between \((3,3)\) and \((6,1)\): \(m_3=\frac{1 - 3}{6 - 3}=\frac{-2}{3}\approx - 0.67\)

Since \(m_1
eq m_3\), Option A is not a linear function.

Step3: Check Option B

• Between \((0,-2)\) and \((2,1)\): \(m_1=\frac{1 - (-2)}{2 - 0}=\frac{3}{2}=1.5\)
• Between \((2,1)\) and \((4,3)\): \(m_2=\frac{3 - 1}{4 - 2}=\frac{2}{2}=1\)

Since \(m_1
eq m_2\), Option B is not a linear function.

Step4: Check Option C

• Between \((-1,6)\) and \((3,0)\): \(m_1=\frac{0 - 6}{3 - (-1)}=\frac{-6}{4}=-\frac{3}{2}=-1.5\)
• Between \((3,0)\) and \((5,-3)\): \(m_2=\frac{-3 - 0}{5 - 3}=\frac{-3}{2}=-1.5\)
• Between \((5,-3)\) and \((9,-9)\): \(m_3=\frac{-9 - (-3)}{9 - 5}=\frac{-6}{4}=-\frac{3}{2}=-1.5\)

All slopes are equal (\(-\frac{3}{2}\)), so the ordered pairs in Option C represent a linear function.

Step5: Check Option D (for completeness)

• Between \((-1,2)\) and \((2,1)\): \(m_1=\frac{1 - 2}{2 - (-1)}=\frac{-1}{3}\approx - 0.33\)
• Between \((2,1)\) and \((4,0)\): \(m_2=\frac{0 - 1}{4 - 2}=\frac{-1}{2}=-0.5\)

Since \(m_1
eq m_2\), Option D is not a linear function.

Answer:

C