16. MODELING WITH MATHEMATICS The table shows the total earnings y (in dollars) of a food server who works $x$ hours. \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $\boldsymbol{x}$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline $\boldsymbol{y}$ & 0 & 18 & 40 & 62 & 77 & 85 & 113 \\ \hline \end{tabular} a. Write an equation that models the server's earnings as a function of the number of hours the server works. b. Interpret the stope and $y$-intercept of the line of fit.

Step 1 ：Observe the data and note that as the number of hours worked increases, the total earnings also increase. This suggests a linear relationship between the number of hours worked (x) and the total earnings (y).

Step 2 ：Calculate the slope of the line using any two points from the table. For example, using the points (1,18) and (2,40), the slope (m) would be \((40-18)/(2-1) = 22\).

Step 3 ：Find the y-intercept, which is the value of y when x is 0. From the table, we can see that when x is 0, y is also 0. So, the y-intercept (b) is 0.

Step 4 ：Write the equation of the line using the slope and the y-intercept. The general form of the equation of a line is y = mx + b. Substituting our values, we get \(y = 22x + 0\), or simply \(y = 22x\).

Step 5 ：Interpret the slope and y-intercept. The slope of the line, 22, represents the rate at which the server's earnings increase for each additional hour worked. The server earns $22 for each hour they work. The y-intercept of the line, 0, represents the server's earnings when they do not work any hours. If the server does not work, they do not earn any money.

Step 6 ：The final equation that models the server's earnings as a function of the number of hours the server works is \(\boxed{y = 22x}\).