To find the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to determine the linear relationship: [tex]\( y = mx + c \)[/tex].
Based on the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, we have:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & -5 \\
\hline
-2 & -1 \\
\hline
-1 & 3 \\
\hline
0 & 7 \\
\hline
1 & 11 \\
\hline
2 & 15 \\
\hline
\end{array}
\][/tex]
First, determine the slope [tex]\( m \)[/tex] and the intercept [tex]\( c \)[/tex]. We have calculated that:
- The slope [tex]\( m \)[/tex] is approximately [tex]\( 4.000000000000002 \)[/tex].
- The intercept [tex]\( c \)[/tex] is [tex]\( 7.0 \)[/tex].
Therefore, the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ y = 4.000000000000002x + 7.0 \][/tex]
Now plugging these values into the table, we get:
[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
-3 & -5 \\
\hline
-2 & -1 \\
\hline
-1 & 3 \\
\hline
0 & 7 \\
\hline
1 & 11 \\
\hline
2 & 15 \\
\hline
\end{tabular}
\][/tex]
So, the final answer for the equation describing the relationship is:
[tex]\[ y = 4x + 7 \][/tex]
Thus, where the table states " [tex]\( y = [?] x + \)[/tex] ", the complete relationship filled in should be:
[tex]\[ y = 4x + 7 \][/tex]
