To find the equation that describes the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex], let's consider the given data points. We observe changes in both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values, and it looks like the relationship between them might be linear. Therefore, we will look for an equation of the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
To find the slope [tex]\(m\)[/tex], let's use the changes in [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. We can calculate the slope as follows:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using any two points from the table, for instance, the first two points [tex]\((-5, 17)\)[/tex] and [tex]\((-4, 14)\)[/tex]:
[tex]\[ \text{Change in } y = 14 - 17 = -3 \][/tex]
[tex]\[ \text{Change in } x = -4 - (-5) = -4 + 5 = 1 \][/tex]
Thus, the slope [tex]\(m\)[/tex] is:
[tex]\[ m = \frac{-3}{1} = -3 \][/tex]
We now have the slope [tex]\(m = -3\)[/tex]. Next, we need to find the y-intercept [tex]\(b\)[/tex]. To do this, we can use one of the points from the table and substitute the values of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(m\)[/tex] into the equation [tex]\(y = mx + b\)[/tex]:
Let's use the point [tex]\((0, 2)\)[/tex]:
[tex]\[ y = mx + b \][/tex]
[tex]\[ 2 = -3(0) + b \][/tex]
[tex]\[ b = 2 \][/tex]
Therefore, the y-intercept [tex]\(b\)[/tex] is [tex]\(2\)[/tex].
Combining the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex], we get the linear equation:
[tex]\[ y = -3x + 2 \][/tex]
So, the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is given by the equation:
[tex]\[ y = -3x + 2 \][/tex]
