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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 [tex]\( a \)[/tex]?

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
10 & 27 \\
\hline
11 & $a$ \\
\hline
12 & 11 \\
\hline
\end{tabular}
\][/tex]

A. [tex]\( a = 2 \)[/tex]
B. [tex]\( a = 3 \)[/tex]
C. [tex]\( a = 19 \)[/tex]
D. [tex]\( a = 35 \)[/tex]



Answer :

GinnyAnswer
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]

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