Alright, let's look at the given data points which represent the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-3 & -5 \\
-2 & -1 \\
-1 & 3 \\
0 & 7 \\
1 & 11 \\
2 & 15 \\
\end{array}
\][/tex]
First, we observe the differences between consecutive [tex]\(y\)[/tex]-values to see if there is a recognizable pattern. Calculate the differences:
[tex]\[
\begin{align*}
-1 - (-5) &= 4, \\
3 - (-1) &= 4, \\
7 - 3 &= 4, \\
11 - 7 &= 4, \\
15 - 11 &= 4.
\end{align*}
\][/tex]
The differences between consecutive [tex]\(y\)[/tex]-values are constant (all equal to 4), suggesting a linear relationship of the form:
[tex]\[
y = mx + c
\][/tex]
Next, we determine the slope ([tex]\(m\)[/tex]) of the line. Since the differences in [tex]\(y\)[/tex]-values are all 4, the slope is:
[tex]\[
m = 4
\][/tex]
Now, we need to find the y-intercept ([tex]\(c\)[/tex]). We use one of the given data points to solve for [tex]\(c\)[/tex]. Let's use the point [tex]\((0, 7)\)[/tex]:
[tex]\[
7 = 4 \cdot 0 + c \implies c = 7
\][/tex]
Thus, the linear equation that describes the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[
y = 4x + 7
\][/tex]
So the completed equation is:
[tex]\[
\boxed{y = 4x + 7}
\][/tex]
