Let's go through the problem step-by-step.
We are given a table of [tex]\( x \)[/tex] values and their corresponding [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\
\hline
y & -1 & -3 & -6 & -8 & -10 & -10 & -10 & -9 & -9 \\
\hline
\end{array}
\][/tex]
Additionally, we are given the equation [tex]\( y = -0.5x - 2 \)[/tex].
Let's verify the given [tex]\( y \)[/tex] values by calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex] using the equation [tex]\( y = -0.5x - 2 \)[/tex].
1. For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -0.5(4) - 2 = -2 - 2 = -4
\][/tex]
2. For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = -0.5(6) - 2 = -3 - 2 = -5
\][/tex]
3. For [tex]\( x = 8 \)[/tex]:
[tex]\[
y = -0.5(8) - 2 = -4 - 2 = -6
\][/tex]
4. For [tex]\( x = 10 \)[/tex]:
[tex]\[
y = -0.5(10) - 2 = -5 - 2 = -7
\][/tex]
5. For [tex]\( x = 12 \)[/tex]:
[tex]\[
y = -0.5(12) - 2 = -6 - 2 = -8
\][/tex]
6. For [tex]\( x = 14 \)[/tex]:
[tex]\[
y = -0.5(14) - 2 = -7 - 2 = -9
\][/tex]
7. For [tex]\( x = 16 \)[/tex]:
[tex]\[
y = -0.5(16) - 2 = -8 - 2 = -10
\][/tex]
8. For [tex]\( x = 18 \)[/tex]:
[tex]\[
y = -0.5(18) - 2 = -9 - 2 = -11
\][/tex]
9. For [tex]\( x = 20 \)[/tex]:
[tex]\[
y = -0.5(20) - 2 = -10 - 2 = -12
\][/tex]
Now, let's compare the calculated [tex]\( y \)[/tex] values with the given [tex]\( y \)[/tex] values:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\
\hline
\text{Given \( y \)} & -1 & -3 & -6 & -8 & -10 & -10 & -10 & -9 & -9 \\
\hline
\text{Calculated \( y \)} & -4 & -5 & -6 & -7 & -8 & -9 & -10 & -11 & -12 \\
\hline
\end{array}
\][/tex]
From the comparison, it's evident that the given [tex]\( y \)[/tex] values do not coincide exactly with the calculated [tex]\( y \)[/tex] values based on the equation [tex]\( y = -0.5x - 2 \)[/tex]. Specifically:
- For [tex]\( x = 4, 6 \)[/tex], the given [tex]\( y \)[/tex] values are higher than the calculated values by 3 and 2, respectively.
- For [tex]\( x = 12 \)[/tex], the given [tex]\( y \)[/tex] value is lower than the calculated value by 2.
- For [tex]\( x = 14, 18, 20 \)[/tex], the given [tex]\( y \)[/tex] values gradually decrease in deviation but still do not match.
Thus, the given [tex]\( y \)[/tex] values do not perfectly follow the equation [tex]\( y = -0.5x - 2 \)[/tex].
