To determine the equation of the line that represents the given data points, we need to find the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]) of the line in the form [tex]\(y = mx + b\)[/tex].
First, let's find the slope ([tex]\(m\)[/tex]) using two points from the table. We'll use the points [tex]\((-2, 16)\)[/tex] and [tex]\((-1, 11)\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the values, we get:
[tex]\[ m = \frac{11 - 16}{-1 - (-2)} = \frac{-5}{1} = -5 \][/tex]
Next, we need to calculate the y-intercept ([tex]\(b\)[/tex]). We can use any of the given points and the slope we just calculated. Let's use the point [tex]\((-2, 16)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
Plugging in the slope and the coordinates of the point, we get:
[tex]\[ 16 = (-5)(-2) + b \][/tex]
[tex]\[ 16 = 10 + b \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ b = 16 - 10 \][/tex]
[tex]\[ b = 6 \][/tex]
Therefore, the equation of the line is:
[tex]\[ y = -5x + 6 \][/tex]
So, the correct equation corresponding to the given data points is [tex]\(y = -5x + 6\)[/tex].
Checking the provided options:
A. [tex]\( y = -5x + 3 \)[/tex]
B. [tex]\( y = -5x + 6 \)[/tex]
C. [tex]\( y = -3x - 5 \)[/tex]
D. [tex]\( y = -5x + 9 \)[/tex]
The correct answer is B: [tex]\(y = -5x + 6\)[/tex].
