To determine the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] based on the given data points, we assume a linear relationship of the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. We will use two points to determine the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex], and then we will verify that the relationship holds for the remaining points.
Given data points:
[tex]\[ (-2, 17), (-1, 12), (0, 7), (1, 2), (2, -3), (3, -8) \][/tex]
First, we select two points to determine [tex]\(m\)[/tex] and [tex]\(b\)[/tex]. Let's use [tex]\((x_1, y_1) = (-2, 17)\)[/tex] and [tex]\((x_2, y_2) = (-1, 12)\)[/tex].
From these two points, we can form two equations:
1. [tex]\( 17 = m(-2) + b \)[/tex]
2. [tex]\( 12 = m(-1) + b \)[/tex]
Now, we solve for [tex]\(m\)[/tex] and [tex]\(b\)[/tex]:
Step 1: Rearrange the first equation:
[tex]\[ 17 = -2m + b \][/tex]
[tex]\[ b = 17 + 2m \tag{Equation 1} \][/tex]
Step 2: Substitute [tex]\(b\)[/tex] from Equation 1 into the second equation:
[tex]\[ 12 = -m + (17 + 2m) \][/tex]
[tex]\[ 12 = 17 + m \][/tex]
[tex]\[ 12 - 17 = m \][/tex]
[tex]\[ m = -5 \][/tex]
Step 3: Substitute [tex]\(m = -5\)[/tex] back into Equation 1 to find [tex]\(b\)[/tex]:
[tex]\[ b = 17 + 2(-5) \][/tex]
[tex]\[ b = 17 - 10 \][/tex]
[tex]\[ b = 7 \][/tex]
Thus, the linear equation describing the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[ y = -5x + 7 \][/tex]
Now, let's verify this relationship using the remaining data points:
For [tex]\(x = -2\)[/tex]:
[tex]\[ y = -5(-2) + 7 = 10 + 7 = 17 \][/tex]
For [tex]\(x = -1\)[/tex]:
[tex]\[ y = -5(-1) + 7 = 5 + 7 = 12 \][/tex]
For [tex]\(x = 0\)[/tex]:
[tex]\[ y = -5(0) + 7 = 0 + 7 = 7 \][/tex]
For [tex]\(x = 1\)[/tex]:
[tex]\[ y = -5(1) + 7 = -5 + 7 = 2 \][/tex]
For [tex]\(x = 2\)[/tex]:
[tex]\[ y = -5(2) + 7 = -10 + 7 = -3 \][/tex]
For [tex]\(x = 3\)[/tex]:
[tex]\[ y = -5(3) + 7 = -15 + 7 = -8 \][/tex]
As we can see, all the points satisfy the equation [tex]\( y = -5x + 7 \)[/tex].
Therefore, the complete equation describing the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[ y = -5x + 7 \][/tex]
