Certainly! Let's tackle this problem step-by-step.
We need to find the slope of the linear relationship between the number of yards [tex]\( y \)[/tex] and the number of feet [tex]\( f \)[/tex], and then interpret this slope in the context of the problem. The table provided gives some conversions between yards and feet:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of yards (} y \text{)} & \text{Number of feet (} f \text{)} \\
\hline
0 & 0 \\
2 & 6 \\
4 & 12 \\
6 & 18 \\
\hline
\end{array}
\][/tex]
Step 1: Understand the points on the graph
Each row from the table represents a point [tex]\((y, f)\)[/tex] on the graph of the line. We have four points here:
- [tex]\((0, 0)\)[/tex]
- [tex]\((2, 6)\)[/tex]
- [tex]\((4, 12)\)[/tex]
- [tex]\((6, 18)\)[/tex]
Step 2: Calculate the slope
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((y_1, f_1)\)[/tex] and [tex]\((y_2, f_2)\)[/tex] is given by:
[tex]\[
m = \frac{f_2 - f_1}{y_2 - y_1}
\][/tex]
We can use the first and the last point from the table for simplicity, which are [tex]\((0, 0)\)[/tex] and [tex]\((6, 18)\)[/tex].
So,
[tex]\[
y_1 = 0, \quad f_1 = 0, \quad y_2 = 6, \quad f_2 = 18
\][/tex]
Now, plug these values into the slope formula:
[tex]\[
m = \frac{18 - 0}{6 - 0} = \frac{18}{6} = 3
\][/tex]
Step 3: Interpretation of the slope
The slope in this context represents the change in feet for each additional yard. Therefore, [tex]\( m = 3 \)[/tex] means that for every 1 yard increase, the number of feet increases by 3.
Hence, the detailed solution is:
[tex]\[
\boxed{3}
\][/tex]
So, there are [tex]\(\boxed{3}\)[/tex] feet per yard.
