To determine the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we need to find the linear equation in the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Given data points:
[tex]\[
(0, 1) \\
(1, -1) \\
(2, -3) \\
(3, -5) \\
(4, -7) \\
(5, -9)
\][/tex]
### Step 1: Calculate the Slope (m)
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the first two points [tex]\((0, 1)\)[/tex] and [tex]\((1, -1)\)[/tex]:
[tex]\[ m = \frac{-1 - 1}{1 - 0} \][/tex]
[tex]\[ m = \frac{-2}{1} \][/tex]
[tex]\[ m = -2 \][/tex]
### Step 2: Calculate the Y-Intercept (b)
The y-intercept [tex]\( b \)[/tex] can be found using the point-slope form of the equation:
[tex]\[ y = mx + b \][/tex]
Using the point [tex]\((0, 1)\)[/tex]:
[tex]\[ 1 = (-2)(0) + b \][/tex]
[tex]\[ 1 = b \][/tex]
Therefore, the y-intercept [tex]\( b \)[/tex] is [tex]\( 1 \)[/tex].
### Step 3: Form the Equation
Now that we have the slope [tex]\( m = -2 \)[/tex] and the y-intercept [tex]\( b = 1 \)[/tex], we can write the equation:
[tex]\[ y = -2x + 1 \][/tex]
So, the completed table with the equation is:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline x & y & y = mx + b \\
\hline 0 & 1 & \\
\hline 1 & -1 & \( y = -2x + 1 \) \\
\hline 2 & -3 & \( y = -2 \cdot x + 1 \) \\
\hline 3 & -5 & \\
\hline 4 & -7 & \\
\hline 5 & -9 & \\
\hline
\end{tabular}
\][/tex]
Thus, the linear equation describing the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ y = -2x + 1 \][/tex]
