Complete the equation describing how [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are related.

[tex]\[
\begin{tabular}{c|c}
$x$ & $y$ \\
\hline
-5 & 17 \\
-4 & 14 \\
-3 & 11 \\
-2 & 8 \\
-1 & 5 \\
0 & 2 \\
\end{tabular}
\][/tex]

[tex]\[
y = [?] x + \square
\][/tex]



Answer :

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]