To fill in the missing values of [tex]\( y \)[/tex] for the given [tex]\( x \)[/tex] values in the table, we need to determine the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Through regression analysis, this relationship can be expressed in the form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
For our data points, the best fit line is given by the equation:
[tex]\[ y = -6.8x + 22.1 \][/tex]
We will use this equation to calculate the values of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex].
1. When [tex]\( x = 3 \)[/tex]:
[tex]\[
y = -6.8 \cdot 3 + 22.1 = -20.4 + 22.1 = 1.7
\][/tex]
2. When [tex]\( x = 6 \)[/tex]:
[tex]\[
y = -6.8 \cdot 6 + 22.1 = -40.8 + 22.1 = -18.7
\][/tex]
3. When [tex]\( x = 9 \)[/tex]:
[tex]\[
y = -6.8 \cdot 9 + 22.1 = -61.2 + 22.1 = -39.1
\][/tex]
4. When [tex]\( x = 10 \)[/tex]:
[tex]\[
y = -6.8 \cdot 10 + 22.1 = -68 + 22.1 = -45.9
\][/tex]
With these calculations, we can fill in the missing values in the table as follows:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
3 & 1.7 \\
\hline
6 & -18.7 \\
\hline
9 & -39.1 \\
\hline
10 & -45.9 \\
\hline
\end{tabular}
\][/tex]
