To derive a rule that best fits the given data points, we can use the method of linear regression. Our goal is to find a linear equation of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Given the data points:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 3 & 6 & 9 & 12 \\
\hline
y & 5 & 14 & 23 & 32 \\
\hline
\end{array}
\][/tex]
Let's find the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex].
### 1. Calculation of the Slope [tex]\( m \)[/tex]
The slope [tex]\( m \)[/tex] of the best-fit line is determined using the following formula:
[tex]\[
m = \frac{n\sum(xy) - \sum(x)\sum(y)}{n\sum(x^2) - (\sum(x))^2}
\][/tex]
Where:
[tex]\[
\begin{aligned}
n &= \text{number of data points} = 4 \\
\sum(x) &= 3 + 6 + 9 + 12 = 30 \\
\sum(y) &= 5 + 14 + 23 + 32 = 74 \\
\sum(xy) &= 3 \cdot 5 + 6 \cdot 14 + 9 \cdot 23 + 12 \cdot 32 = 15 + 84 + 207 + 384 = 690 \\
\sum(x^2) &= 3^2 + 6^2 + 9^2 + 12^2 = 9 + 36 + 81 + 144 = 270
\end{aligned}
\][/tex]
Plugging in these values:
[tex]\[
m = \frac{4 \cdot 690 - 30 \cdot 74}{4 \cdot 270 - 30^2} = \frac{2760 - 2220}{1080 - 900} = \frac{540}{180} = 3
\][/tex]
### 2. Calculation of the Intercept [tex]\( b \)[/tex]
The intercept [tex]\( b \)[/tex] can be calculated using the formula:
[tex]\[
b = \frac{\sum(y) - m\sum(x)}{n}
\][/tex]
Plugging in the values and the slope:
[tex]\[
b = \frac{74 - 3 \cdot 30}{4} = \frac{74 - 90}{4} = \frac{-16}{4} = -4
\][/tex]
So the intercept [tex]\( b \)[/tex] is -4.
### 3. Forming the Linear Equation
The best-fit linear equation for the data is:
[tex]\[
y = 3x - 4
\][/tex]
### Conclusion
The rule that best fits the data in the table is:
[tex]\[
\boxed{y = 3x - 4}
\][/tex]
