To find the linear equation that defines the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the table, we need to determine the slope and the y-intercept.
1. Determine the Slope (m):
The slope [tex]\( m \)[/tex] is calculated using two points from the table. Let's use the points (36, 8) and (41, 18):
[tex]\[
m = \frac{y_1 - y_0}{x_1 - x_0} = \frac{18 - 8}{41 - 36} = \frac{10}{5} = 2.0
\][/tex]
2. Determine the y-intercept (b):
The y-intercept [tex]\( b \)[/tex] can be found using the formula [tex]\( y = mx + b \)[/tex]. Using the point (36, 8) and the slope [tex]\( m = 2.0 \)[/tex]:
[tex]\[
8 = 2.0 \cdot 36 + b \implies 8 = 72 + b \implies b = 8 - 72 = -64.0
\][/tex]
3. Form the Equation:
With the slope [tex]\( m = 2.0 \)[/tex] and y-intercept [tex]\( b = -64.0 \)[/tex], the linear equation is:
[tex]\[
y = 2.0x - 64.0
\][/tex]
Therefore, the linear equation that defines the rule for the given table is:
[tex]\[
y = 2.0x - 64.0
\][/tex]
