Question

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 $a$?

$x$	$y$
11	$a$
12	11

options: $a = 2$, $a = 35$, $a = 3$, $a = 19$

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). For a linear function, this rate is constant. Here, rate of change is \(-8\), and we can use points \((10, 27)\) and \((11, a)\), or \((11, a)\) and \((12, 11)\). Let's use \((10, 27)\) and \((11, a)\). The change in \(x\) is \(11 - 10 = 1\), change in \(y\) is \(a - 27\). So rate of change is \(\frac{a - 27}{11 - 10}=-8\).

Step2: Solve for \(a\)

Simplify the equation: \(\frac{a - 27}{1}=-8\) (since \(11 - 10 = 1\)). So \(a - 27=-8\). Add 27 to both sides: \(a=-8 + 27\). Calculate: \(a = 19\). We can check with the next pair \((11, 19)\) and \((12, 11)\). Rate of change is \(\frac{11 - 19}{12 - 11}=\frac{-8}{1}=-8\), which matches.

Answer:

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