Irma owns a cake shop and she is working on two wedding cakes this week. The first cake consists of 4 small tiers and 3 large tiers, which will serve a total of 236 guests. The second one includes 1 small tier and 5 large tiers, which is enough servings for 280 guests. How many guests does each size of tier serve?
Let's assume that the number of guests served by a small tier is s, and the number of guests served by a large tier is l.
From the information given, we can form two equations:
4s + 3l = 236 -- equation (1)
1s + 5l = 280 -- equation (2)
Let's solve equation (1) first:
Multiply equation (1) by 5, which gives:
20s + 15l = 1180 -- equation (3)
Now, multiply equation (2) by 4, which gives:
4s + 20l = 1120 -- equation (4)
Subtract equation (4) from equation (3):
(20s + 15l) - (4s + 20l) = 1180 - 1120
16s - 5l = 60 -- equation (5)
Now, we can solve equations (5) and (2) simultaneously.
Multiply equation (5) by 5, and subtract it from equation (2):
5s + 25l - (16s - 5l) = 280 - 300
5s + 25l - 16s + 5l = -20
-11s + 30l = -20 -- equation (6)
Now, multiply equation (5) by 3, and subtract it from equation (1):
(20s + 15l) - (3s + 9l) = 1180 - 354
17s + 6l = 826 -- equation (7)
We have a system of equations:
-11s + 30l = -20 -- equation (6)
17s + 6l = 826 -- equation (7)
By solving these equations, we find that each small tier serves 14 guests, and each large tier serves 36 guests.