To determine which equation represents the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given in the table, we need to find the slope of the line (the rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]) and the y-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]).
We have the following data points:
- Point 1: [tex]\((x_1, y_1) = (0, 18)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (1, 13)\)[/tex]
- Point 3: [tex]\((x_3, y_3) = (2, 8)\)[/tex]
First, we calculate the slope [tex]\( m \)[/tex]. The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using Points 1 and 2:
[tex]\[ m = \frac{13 - 18}{1 - 0} \][/tex]
[tex]\[ m = \frac{-5}{1} \][/tex]
[tex]\[ m = -5 \][/tex]
Thus, the slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex].
Next, we need to find the y-intercept [tex]\( b \)[/tex]. We can use any of the given points and the slope-intercept form of the equation of a line:
[tex]\[ y = mx + b \][/tex]
Using Point 1 [tex]\((0, 18)\)[/tex]:
[tex]\[ 18 = -5 \cdot 0 + b \][/tex]
[tex]\[ 18 = b \][/tex]
Thus, the y-intercept [tex]\( b \)[/tex] is [tex]\( 18 \)[/tex].
Combining the slope and the y-intercept, the equation of the line is:
[tex]\[ y = -5x + 18 \][/tex]
Therefore, the correct equation that represents the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ \boxed{y = -5x + 18} \][/tex]
Option (D) is the correct answer.
