Certainly! Let's determine the equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] using the given data points. The table provided is as follows:
[tex]\[
\begin{tabular}{c|c}
$x$ & $y$ \\
\hline
0 & 1 \\
1 & -1 \\
2 & -3 \\
3 & -5 \\
4 & -7 \\
5 & -9 \\
\end{tabular}
\][/tex]
First, observe the pattern in the [tex]\( y \)[/tex]-values as [tex]\( x \)[/tex] increases.
To find the pattern:
- Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
\begin{align*}
y(1) - y(0) &= -1 - 1 = -2, \\
y(2) - y(1) &= -3 - (-1) = -2, \\
y(3) - y(2) &= -5 - (-3) = -2, \\
y(4) - y(3) &= -7 - (-5) = -2, \\
y(5) - y(4) &= -9 - (-7) = -2.
\end{align*}
\][/tex]
The differences are consistent and equal to [tex]\(-2\)[/tex]. This indicates that the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is linear and can be written in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
From our differences:
[tex]\[ m = -2 \][/tex]
Next, let's find [tex]\( b \)[/tex], the y-intercept. We recognize that the y-intercept [tex]\( b \)[/tex] is simply the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]:
[tex]\[ y(0) = 1 \][/tex]
Thus,
[tex]\[ b = 1 \][/tex]
Therefore, the equation relating [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
y = -2x + 1
\][/tex]
So, to summarize:
[tex]\[
y = -2x + 1
\][/tex]
Here, the slope ([tex]\(-2\)[/tex]) is replaced in the place of the question mark [tex]\([?]\)[/tex], and the y-intercept ([tex]\(1\)[/tex]) replaces the square ([tex]\(\square\)[/tex]) symbol.
Thus, the complete equation describing the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
y = -2x + 1
\][/tex]
