To find the linear equation that describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given table, we'll find the slope of the line and the y-intercept.
The table provides the following data points:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline 2 & 91 \\
\hline 3 & 94 \\
\hline 4 & 97 \\
\hline 5 & 100 \\
\hline
\end{array}
\][/tex]
1. Calculate the slope [tex]\( m \)[/tex]:
The slope formula is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the points (2, 91) and (5, 100), we have:
[tex]\[
m = \frac{100 - 91}{5 - 2} = \frac{9}{3} = 3.0
\][/tex]
2. Calculate the y-intercept [tex]\( b \)[/tex]:
Using the slope [tex]\( m = 3.0 \)[/tex] and one of the points, such as (2, 91), we use the formula for calculating the y-intercept:
[tex]\[
y = mx + b \implies 91 = 3.0 \cdot 2 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
91 = 6 + b \implies b = 91 - 6 = 85.0
\][/tex]
3. Form the linear equation:
The linear equation in the form [tex]\( y = mx + b \)[/tex] is:
[tex]\[
y = 3.0x + 85.0
\][/tex]
So, the rule for this table is given by the equation:
[tex]\[
y = 3.0x + 85.0
\][/tex]
