To determine the equation of the line of best fit for the data that consists of the weeks and the corresponding miles Rita runs, we need to find the linear relationship between the two variables.
In a linear equation of the form [tex]\( y = mx + b \)[/tex], the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] define the line. Here, [tex]\( x \)[/tex] represents the number of weeks, and [tex]\( y \)[/tex] represents the miles run.
Step-by-Step Process:
1. Identify the Data Points:
The given data is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Week (x)} & \text{Miles Run (y)} \\
\hline
1 & 5 \\
\hline
2 & 8 \\
\hline
4 & 13 \\
\hline
6 & 15 \\
\hline
8 & 19 \\
\hline
10 & 20 \\
\hline
\end{array}
\][/tex]
2. Calculate the Slope (m):
The slope of the line measures how much [tex]\( y \)[/tex] changes for a one-unit change in [tex]\( x \)[/tex].
3. Determine the Y-Intercept (b):
The y-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
The calculations yield the following values for the slope and intercept:
- The slope [tex]\( m \approx 1.671 \)[/tex]
- The y-intercept [tex]\( b \approx 4.699 \)[/tex]
4. Form the Equation:
Substitute these values into the linear equation [tex]\( y = mx + b \)[/tex]:
[tex]\[
y \approx 1.671x + 4.699
\][/tex]
So, the equation of the line of best fit for Rita's running data, rounded to the thousandths place, is:
[tex]\[ y \approx 1.671x + 4.699 \][/tex]
Thus, the equation that represents the line of best fit for the given data is:
[tex]\[ y \approx 1.671x + 4.699 \][/tex]
