A company sells 3 items for 3 dollars and 12 items for 10 dollars. If it is a linear model, find the equation of the line that models the situation. Graph the line. How much should that company charge for selling a single item? How much should you charge for selling 10 items? How much should you charge for selling 18 items? How would you explain the company charging when the customer purchases zero items?

To find the equation of the line that models the situation, we can use the following two points: ``` (3, 3) (12, 10) ``` The slope of the line is: ``` m = (10 - 3) / (12 - 3) = 7 / 9 ``` The y-intercept of the line is: ``` b = 3 - 7/9 * 3 = 2/9 ``` Therefore, the equation of the line is: ``` y = 7/9 * x + 2/9 ``` **Graph of the line:** [asy] unitsize(1 cm); pair A, B; A = (0,2/9); B = (9,8); draw(graph(A,B),red); label("$(3, 3)$", (3,3), SE); label("$(12, 10)$", (12,10), SE); label("$(0, 2/9)$", (0,2/9), S); label("$(9, 8)$", (9,8), NE); label("$y$", (10,0), S); label("$x$", (0,1), W); [/asy] **How much should that company charge for selling a single item?** To calculate the price for selling a single item, we can substitute x = 1 into the equation of the line: ``` y = 7/9 * 1 + 2/9 = 1 ``` Therefore, the company should charge **1 dollar** for selling a single item. **How much should you charge for selling 10 items?** To calculate the price for selling 10 items, we can substitute x = 10 into the equation of the line: ``` y = 7/9 * 10 + 2/9 = 7.8 ``` Therefore, the company should charge **7.8 dollars** for selling 10 items. **How much should you charge for selling 18 items?** To calculate the price for selling 18 items, we can substitute x = 18 into the equation of the line: ``` y = 7/9 * 18 + 2/9 = 14.8 ``` Therefore, the company should charge **14.8 dollars** for selling 18 items. **How would you explain the company charging when the customer purchases zero items?** The company charging a fee for selling zero items is a common practice in some industries. It is known as a "convenience fee" or a "processing fee." The company may charge this fee to cover the costs of processing the transaction, even if the customer is not purchasing any goods or services. In some cases, the company may also charge a fee for selling zero items in order to discourage customers from returning items. For example, a company that sells clothing may charge a fee for returning items. This fee is designed to discourage customers from buying clothes, trying them on, and then returning them if they don't fit. Whether or not it is fair for a company to charge a fee for selling zero items is a matter of opinion. Some people believe that it is unfair to charge customers for a service that they are not receiving. Others believe that it is a fair way for the company to cover its costs. In the case of the company selling items for 3 dollars and 12 dollars, it is possible that the company is charging a fee for selling zero items to cover the costs of processing the transaction. It is also possible that the company is charging this fee to discourage customers from returning items.