To determine the linear equation that represents the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we need to follow a few steps.
First, observe that as [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] also increases by 1. This indicates a linear relationship.
To find the equation of a line, we use the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's determine the slope [tex]\( m \)[/tex]:
We can use any two points from the table to find the slope. For example, using the points (5, 31) and (6, 32):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{32 - 31}{6 - 5} = \frac{1}{1} = 1
\][/tex]
So, the slope [tex]\( m \)[/tex] is 1.
Next, we need to find the y-intercept [tex]\( b \)[/tex]. We can use one of the points from the table and substitute it into the linear equation [tex]\( y = mx + b \)[/tex]. Let's use the point (5, 31):
[tex]\[
31 = 1 \cdot 5 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
31 = 5 + b \implies b = 31 - 5 \implies b = 26
\][/tex]
So, the y-intercept [tex]\( b \)[/tex] is 26.
Therefore, the linear equation that gives the rule for this table is:
[tex]\[
\boxed{y = 1x + 26}
\][/tex]
