Emilia and Jessica are hosting events that are catered by the same company. Emilia plans to have 96 adults and 42 children attend, so the total projected cost of her meals is $3,336. Jessica has 79 adults and 74 children on her guest list, so she will pay the caterer $3,534. How much does the caterer charge for an adult meal and for an child's meal?
Let A be the cost of an adult meal and C be the cost of a child's meal.
From the problem, we have the following two equations:
96A + 42C = 3336 (1)
79A + 74C = 3534 (2)
To solve this system of equations, we can multiply equation (2) by 2, then subtract equation (1) from equation (2):
158A + 148C - (96A + 42C) = 3534 × 2 - 3336
62A + 106C = 7064
Next, we can multiply equation (1) by 2 and then subtract equation (2) from equation (1):
192A + 84C - (79A + 74C) = 3336 × 2 - 3534
113A + 10C = 3204
Now we have the following system of equations:
62A + 106C = 7064 (3)
113A + 10C = 3204 (4)
We can use the elimination method to solve this system of equations:
Multiply equation (4) by 6 and equation (3) by 10, then subtract equation (4) from equation (3):
620A + 1060C - (678A + 60C) = 3204 × 10 - 113 × 6
-58A + 1000C = 18804 - 678
-58A + 1000C = 18126
Divide both sides by 2:
-29A + 500C = 9063 (5)
Multiply equation (4) by 5 and equation (3) by 11, then subtract equation (3) from equation (4):
565A + 55C - (682A + 110C) = 113 × 11 - 7064 × 5
-117A - 55C = -74049 - 35320
-117A - 55C = -109369
Divide both sides by -2:
59A + 27.5C = 54684.5 (6)
Add equations (5) and (6) together:
-29A + 500C + 59A + 27.5C = 9063 + 54684.5
30A + 527.5C = 63747.5
Divide both sides by 5:
6A + 105.5C = 12749.5 (7)
Multiply equation (6) by 4 and equation (5) by 5, then subtract equation (6) from equation (5):
290A + 110C - (118A + 110C) = 54516 × 5 - 63747.5 × 4
172A = 109080 - 254990
172A = -145910
Divide both sides by 172:
A = -145910/172
A = -848.32
Substituting the value of A into equation (4):
113 * -848.32 + 10C = 3204
-95857.6 + 10C = 3204
10C = 99061.6
C = 99061.6/10
C = 9906.16
The caterer charges $848.32 for an adult meal and $9906.16 for a child's meal.