To find the equation that describes the relationship between \( x \) and \( y \) for the given values, we can determine the linear relationship by finding the slope (\( m \)) and the y-intercept (\( b \)) using the least squares method.
Given data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 3 & 6 & 9 & 12 & 15 \\
\hline
y & \frac{17}{7} & \frac{32}{7} & \frac{47}{7} & \frac{62}{7} & \frac{77}{7} \\
\hline
\end{array}
\][/tex]
1. Calculate the slope \( m \):
The slope (\( m \)) is calculated by determining the rate of change in \( y \) with respect to \( x \).
2. Calculate the y-intercept \( b \):
The y-intercept (\( b \)) is found where the line intersects the y-axis.
After performing the necessary computations, the results are:
- The slope \( m = 0.7142857142857144 \)
- The y-intercept \( b = 0.2857142857142819 \)
3. Formulate the equation of the line:
The equation of a line in its general form is:
[tex]\[
y = mx + b
\][/tex]
Substituting the values of \( m \) and \( b \) into this equation, we get:
[tex]\[
y = 0.7142857142857144x + 0.2857142857142819
\][/tex]
Thus, the equation that shows the relationship between \( x \) and \( y \) based on the given table of values is:
[tex]\[
y = 0.7142857142857144x + 0.2857142857142819
\][/tex]
