To determine the type of relationship between the given [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values, we will analyze the data points step-by-step.
### Given Table of Values
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & -2 \\
\hline
6 & 9 \\
\hline
7 & 20 \\
\hline
8 & 31 \\
\hline
9 & 42 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Analysis
#### Step 1: Check for Linearity
To check if the relationship is linear, we compute the first differences between successive [tex]\(y\)[/tex] values.
[tex]\[
\text{First differences:}
\][/tex]
[tex]\[
\begin{align*}
y_6 - y_5 &= 9 - (-2) = 11, \\
y_7 - y_6 &= 20 - 9 = 11, \\
y_8 - y_7 &= 31 - 20 = 11, \\
y_9 - y_8 &= 42 - 31 = 11.
\end{align*}
\][/tex]
If all the differences are equal, the data points form a linear relationship. Here, the first differences are:
[tex]\[
11, 11, 11, 11.
\][/tex]
Since all the first differences are the same, the data suggests a linear relationship.
Thus, we conclude:
The relationship is linear.
Given the options:
- Linear
- Quadratic
- Exponential
- None of the above
### Final Answer
[tex]\[
\boxed{1 \text{ or Linear}}
\][/tex]
This analysis shows that the relationship between the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values in the table is linear.
