Question

which set of ordered pairs is linear?
a. {(-2, 3), (-1, 6), (0, 12)}
b. {(3, 1), (5, 2), (10, 10)}
c. {(-1, 0), (2, 3), (5, 6)}

Explanation:

Step1: Recall linearity test

A set of points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) is linear if the slope between any two pairs is equal. The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Check Option A

Points: \((-2,3)\), \((-1,8)\), \((0,13)\)
Slope between \((-2,3)\) and \((-1,8)\): \(m_1=\frac{8 - 3}{-1-(-2)}=\frac{5}{1}=5\)
Slope between \((-1,8)\) and \((0,13)\): \(m_2=\frac{13 - 8}{0 - (-1)}=\frac{5}{1}=5\)
Slopes are equal, so check if linear.

Step3: Check Option B

Points: \((3,1)\), \((5,2)\), \((10,10)\)
Slope between \((3,1)\) and \((5,2)\): \(m_1=\frac{2 - 1}{5 - 3}=\frac{1}{2}=0.5\)
Slope between \((5,2)\) and \((10,10)\): \(m_2=\frac{10 - 2}{10 - 5}=\frac{8}{5}=1.6\)
Slopes are not equal, so not linear.

Step4: Check Option C

Points: \((-2,0)\), \((2,3)\), \((5,6)\)
Slope between \((-2,0)\) and \((2,3)\): \(m_1=\frac{3 - 0}{2 - (-2)}=\frac{3}{4}=0.75\)
Slope between \((2,3)\) and \((5,6)\): \(m_2=\frac{6 - 3}{5 - 2}=\frac{3}{3}=1\)
Slopes are not equal, so not linear.

Answer:

A. \((-2, 3)\), \((-1, 8)\), \((0, 13)\)