Question

which set of ordered pairs ((x, y)) could represent a linear function?

( mathbf{a} = {(-8, 8), (-4, 5), (0, 2), (5, -1)} )

( mathbf{b} = {(-1, -2), (2, 1), (5, 4), (8, 7)} )

( mathbf{c} = {(-2, 8), (1, 5), (4, 2), (6, -1)} )

( mathbf{d} = {(-1, 0), (1, 1), (7, 5), (9, 6)} )

answer
( circ ) a ( circ ) b ( circ ) c ( circ ) d

Explanation:

Step1: Recall slope formula

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

Step2: Calculate slope for set A

• Between \((-8,8)\) and \((-4,5)\): \(m_1=\frac{5 - 8}{-4 - (-8)}=\frac{-3}{4}=-\frac{3}{4}\)
• Between \((-4,5)\) and \((0,2)\): \(m_2=\frac{2 - 5}{0 - (-4)}=\frac{-3}{4}=-\frac{3}{4}\)
• Between \((0,2)\) and \((5,-1)\): \(m_3=\frac{-1 - 2}{5 - 0}=\frac{-3}{5}=-0.6\)

Slopes are not equal, so A is not linear.

Step3: Calculate slope for set B

• Between \((-1,-2)\) and \((2,1)\): \(m_1=\frac{1 - (-2)}{2 - (-1)}=\frac{3}{3}=1\)
• Between \((2,1)\) and \((5,4)\): \(m_2=\frac{4 - 1}{5 - 2}=\frac{3}{3}=1\)
• Between \((5,4)\) and \((8,7)\): \(m_3=\frac{7 - 4}{8 - 5}=\frac{3}{3}=1\)

Slopes are equal (all 1), so check if it's linear.

Step4: Calculate slope for set C (optional, but for completeness)

• Between \((-2,8)\) and \((1,5)\): \(m_1=\frac{5 - 8}{1 - (-2)}=\frac{-3}{3}=-1\)
• Between \((1,5)\) and \((4,2)\): \(m_2=\frac{2 - 5}{4 - 1}=\frac{-3}{3}=-1\)
• Between \((4,2)\) and \((6,-1)\): \(m_3=\frac{-1 - 2}{6 - 4}=\frac{-3}{2}=-1.5\)

Slopes are not equal, so C is not linear.

Step5: Calculate slope for set D (optional, but for completeness)

• Between \((-1,0)\) and \((1,1)\): \(m_1=\frac{1 - 0}{1 - (-1)}=\frac{1}{2}=0.5\)
• Between \((1,1)\) and \((7,5)\): \(m_2=\frac{5 - 1}{7 - 1}=\frac{4}{6}=\frac{2}{3}\approx0.666\)

Slopes are not equal, so D is not linear.

Answer:

B. \(\{(-1, -2), (2, 1), (5, 4), (8, 7)\}\)