To determine the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we can observe the provided data points:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-3 & -5 \\
\hline
-2 & -3 \\
\hline
-1 & -1 \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 5 \\
\hline
\end{tabular}
\][/tex]
We need to find the pattern or a mathematical formula that relates [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. By examining the data:
1. When [tex]\(x = -3\)[/tex], [tex]\(y = -5\)[/tex]
2. When [tex]\(x = -2\)[/tex], [tex]\(y = -3\)[/tex]
3. When [tex]\(x = -1\)[/tex], [tex]\(y = -1\)[/tex]
4. When [tex]\(x = 0\)[/tex], [tex]\(y = 1\)[/tex]
5. When [tex]\(x = 1\)[/tex], [tex]\(y = 3\)[/tex]
6. When [tex]\(x = 2\)[/tex], [tex]\(y = 5\)[/tex]
It looks like [tex]\(y\)[/tex] increases linearly as [tex]\(x\)[/tex] increases. Let's propose a linear relationship of the form [tex]\( y = mx + c \)[/tex] where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
To find the slope [tex]\(m\)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\((x_1, y_1) = (-3, -5)\)[/tex] and [tex]\((x_2, y_2) = (-2, -3)\)[/tex]:
[tex]\[ m = \frac{-3 - (-5)}{-2 - (-3)} = \frac{-3 + 5}{-2 + 3} = \frac{2}{1} = 2 \][/tex]
Now, use the slope [tex]\(m = 2\)[/tex] to find the y-intercept [tex]\(c\)[/tex]. Substitute one of the points, say [tex]\((x, y) = (0, 1)\)[/tex], into the linear equation [tex]\(y = mx + c\)[/tex]:
[tex]\[ 1 = 2(0) + c \][/tex]
[tex]\[ c = 1 \][/tex]
Therefore, the equation that describes the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[ y = 2x + 1 \][/tex]
So, the final answer for the equation relating [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is:
[tex]\[ y = 2x + 1 \][/tex]
