To determine the value of [tex]\(a\)[/tex] such that the data in the table represents a linear function with a rate of change (slope) of [tex]\(-8\)[/tex], we need to use the slope formula.
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on a line is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given the coordinates [tex]\((10, 27)\)[/tex], [tex]\((11, a)\)[/tex], and [tex]\((12, 11)\)[/tex], we should check the slope between these points to ensure it is consistent with the given rate of change of [tex]\(-8\)[/tex].
First, let's calculate the slope between [tex]\((10, 27)\)[/tex] and [tex]\((11, a)\)[/tex]:
[tex]\[ m = \frac{a - 27}{11 - 10} \][/tex]
Since [tex]\(11 - 10 = 1\)[/tex], this simplifies to:
[tex]\[ m = a - 27 \][/tex]
We know the slope should be [tex]\(-8\)[/tex]:
[tex]\[ a - 27 = -8 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a - 27 = -8 \][/tex]
[tex]\[ a = 27 - 8 \][/tex]
[tex]\[ a = 19 \][/tex]
Thus, the value of [tex]\(a\)[/tex] must be [tex]\(19\)[/tex]. Therefore, the correct answer is:
[tex]\[ a = 19 \][/tex]
