To solve the problem, let's analyze the information given and determine which points lie on the line representing the relationship between the number of diners and the bill amounts.
The table given shows the average bill totals for a certain number of diners:
[tex]\[
\begin{tabular}{|c|c|}
\hline Persons & Bill Amount \\
\hline 1 & \$ 40 \\
\hline 2 & \$ 80 \\
\hline 7 & \$ 280 \\
\hline 9 & \$ 360 \\
\hline 11 & \$ 440 \\
\hline
\end{tabular}
\][/tex]
First, we determine the rate (slope) of the line. Considering the points (1, 40) and (2, 80):
[tex]\[ \text{Rate} = \frac{80 - 40}{2 - 1} = \frac{40}{1} = 40.0 \][/tex]
Next, we determine the y-intercept using one of the points, say (1, 40):
[tex]\[ \text{y-intercept} = 40 - (40 \times 1) = 0.0 \][/tex]
Now, we have the line equation for this relationship:
[tex]\[ y = 40x + 0 \][/tex]
[tex]\[ y = 40x \][/tex]
Using this relationship, check each given point (x, y):
1. For [tex]\((12, 460)\)[/tex]:
[tex]\[ y = 40 \times 12 = 480 \][/tex]
[tex]\[460 \neq 480\][/tex]
2. For [tex]\((20, 800)\)[/tex]:
[tex]\[ y = 40 \times 20 = 800 \][/tex]
[tex]\[800 = 800\][/tex]
3. For [tex]\((24, 1020)\)[/tex]:
[tex]\[ y = 40 \times 24 = 960 \][/tex]
[tex]\[1020 \neq 960\][/tex]
4. For [tex]\((27, 1080)\)[/tex]:
[tex]\[ y = 40 \times 27 = 1080 \][/tex]
[tex]\[1080 = 1080\][/tex]
Based on these calculations, the points that lie on the line represented by the equation [tex]\(y = 40x\)[/tex] are:
[tex]\[
(20, 800) \\
(27, 1080)
\][/tex]
Thus, the correct answers are:
[tex]\[
(20, 800) \\
(27, 1080)
\][/tex]
