To find the equation of the linear relationship represented by the table, we need to determine two key components: the slope (m) and the y-intercept (b). The slope-intercept form of a linear equation is given by [tex]\( y = mx + b \)[/tex].
1. Identify the points from the table:
[tex]\[
\begin{aligned}
(x_1, y_1) &= (0, -6) \\
(x_2, y_2) &= (1, -3) \\
(x_3, y_3) &= (2, 0) \\
(x_4, y_4) &= (3, 3) \\
\end{aligned}
\][/tex]
2. Calculate the slope (m):
The slope [tex]\( m \)[/tex] can be calculated using any two points, but typically we use the formula:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values from the first two points:
[tex]\[
\Delta y = -3 - (-6) = 3
\][/tex]
[tex]\[
\Delta x = 1 - 0 = 1
\][/tex]
Therefore,
[tex]\[
m = \frac{3}{1} = 3
\][/tex]
3. Find the y-intercept (b):
Using the slope-intercept form [tex]\( y = mx + b \)[/tex], we can plug in the coordinates of any point to solve for [tex]\( b \)[/tex]. Using the point [tex]\( (0, -6) \)[/tex]:
[tex]\[
-6 = 3(0) + b
\][/tex]
Therefore,
[tex]\[
b = -6
\][/tex]
4. Write the equation:
With [tex]\( m = 3 \)[/tex] and [tex]\( b = -6 \)[/tex], the equation of the linear relationship is:
[tex]\[
y = 3x - 6
\][/tex]
Thus, the correct equation of the linear relationship based on the table is:
[tex]\[
y = 3x - 6
\][/tex]
The correct choice is:
[tex]\( \boxed{y = 3x - 6} \)[/tex]
