Question

x | y
10 | 27
11 | a
12 | 11
in order for the data in the table to represent a linear function with a rate of change of -8, what must be the value of a?
○ a = 2
○ a = 3
○ a = 19
○ a = 35

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, rate of change \( m=-8 \). Let's take points \((10, 27)\) and \((11, a)\) first. The change in \( x \) is \( 11 - 10 = 1 \), change in \( y \) is \( a - 27 \). So \( -8=\frac{a - 27}{11 - 10} \).

Step2: Solve for \( a \)

Simplify the equation: \( -8=\frac{a - 27}{1} \), which means \( a - 27=-8 \). Add 27 to both sides: \( a=-8 + 27 = 19 \). We can also check with points \((11, 19)\) and \((12, 11)\): \( \frac{11 - 19}{12 - 11}=\frac{-8}{1}=-8 \), which matches the rate of change.

Answer:

\( a = 19 \) (corresponding to the option: \( a = 19 \))