Question

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

Take points \((-8,9)\) and \((-3,4)\): \(m_1=\frac{4 - 9}{-3 - (-8)}=\frac{-5}{5}=-1\)
Take points \((-3,4)\) and \((1,-1)\): \(m_2=\frac{-1 - 4}{1 - (-3)}=\frac{-5}{4}=-1.25\)
Slopes are not equal, so A is not linear.

Step3: Calculate slope for set B

Take points \((1,-8)\) and \((3,-5)\): \(m_1=\frac{-5 - (-8)}{3 - 1}=\frac{3}{2}=1.5\)
Take points \((3,-5)\) and \((5,-2)\): \(m_2=\frac{-2 - (-5)}{5 - 3}=\frac{3}{2}=1.5\)
Take points \((5,-2)\) and \((7,1)\): \(m_3=\frac{1 - (-2)}{7 - 5}=\frac{3}{2}=1.5\)
All slopes are equal (\(1.5\)), so B has constant slope.

Step4: Verify set C (optional, but for thoroughness)

Take points \((-2,9)\) and \((0,4)\): \(m_1=\frac{4 - 9}{0 - (-2)}=\frac{-5}{2}=-2.5\)
Take points \((0,4)\) and \((2,0)\): \(m_2=\frac{0 - 4}{2 - 0}=\frac{-4}{2}=-2\)
Slopes differ, so C is not linear.

Step5: Verify set D (optional)

Take points \((-2,-3)\) and \((1,0)\): \(m_1=\frac{0 - (-3)}{1 - (-2)}=\frac{3}{3}=1\)
Take points \((1,0)\) and \((5,3)\): \(m_2=\frac{3 - 0}{5 - 1}=\frac{3}{4}=0.75\)
Slopes differ, so D is not linear.

Answer:

B. \(\{(1, -8), (3, -5), (5, -2), (7, 1)\}\)