1)) Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. ()) A charity is considering the possibility of having a benefit night at two different restaurants. The owner of the local Italian restaurant has offered to make a donation of \( \$ 234 \) and \( \$ 1 \) per diner that night. On the other hand, the owner of the Mexican restaurant has said he could contribute \( \$ 206 \) plus \( \$ 3 \) per diner. Based on the number of diners who have promised to participate in the event; it appears that each restaurant would donate the same total amount. How many diners promised to participate? How much would each restaurant donate? ()) With

Let \( x \) represent the number of diners at the Italian restaurant. The total donation from the Italian restaurant can be expressed as: \[ 234 + 1x \] For the Mexican restaurant, with \( y \) being the number of diners, the total donation can be expressed as: \[ 206 + 3y \] Since the problem states that both donations are equal, we set up the equation: \[ 234 + 1x = 206 + 3y \] To satisfy the condition, we know the number of diners promised at both restaurants will be equal, so \( x = y \). Now we can rewrite our equation: \[ 234 + 1x = 206 + 3x \] This simplifies to: \[ 234 - 206 = 3x - x \] \[ 28 = 2x \] \[ x = 14 \] Thus, there are 14 diners promised to participate at each restaurant. Now, substituting \( x \) back to find the donation amount: Italian restaurant donation: \[ 234 + 1(14) = 234 + 14 = 248 \] Mexican restaurant donation: \[ 206 + 3(14) = 206 + 42 = 248 \] To summarize, 14 diners promised to participate, and each restaurant would donate \( \$ 248 \). --- The concept of benefit nights has a charming history steeped in community spirit! The first fundraisers typically took place in the 19th century, where gatherings at local establishments created a sense of camaraderie while raising funds for local causes—talk about dining for a cause! When planning your own benefit night, it's essential to budget wisely and understand the expected number of participants. A common mistake is underestimating the expenses that could arise from hosting an event, such as venue costs and promotional materials. Planning carefully helps ensure that the event is profitable for your charity!