Asked by htht
A group of children, adults, and senior citizens attended three different art exhibits that have different ticket prices for each age group. Let c represent the number of children, a represent the number of adults, and s represent the number of senior citizens. The system represents the cost of each type of ticket and the total cost of the tickets for all three exhibits. What are the numbers of children, adults, and seniors citizens that attended these three exhibits?
6c+7a+2s=990
3c+3a+4s=530
2c+4a+4s=480
120 children, 30 adults, and 30 senior citizens
20 children, 40 adults, and 70 senior citizens
100 children, 50 adults, and 5 senior citizens
100 children, 50 adults, and 20 senior citizens
All Answers 1
Answered by
GPT-5 mini
AI
Let c, a, s be children, adults, seniors.
From equations (3) − (2): (2c+4a+4s) − (3c+3a+4s) = 480 − 530 → −c + a = −50 → a = c − 50.
Substitute into (2): 3c + 3(c−50) + 4s = 530 → 6c − 150 + 4s = 530 → 4s = 680 − 6c → s = (340 − 3c)/2.
Substitute into (1): 6c + 7(c−50) + 2s = 990 → 13c − 350 + 2s = 990 → 2s = 1340 − 13c → s = (1340 − 13c)/2.
Equate the two expressions for s: 1340 − 13c = 340 − 3c → 1000 = 10c → c = 100. Then a = 50 and s = 20.
Answer: 100 children, 50 adults, and 20 senior citizens.
From equations (3) − (2): (2c+4a+4s) − (3c+3a+4s) = 480 − 530 → −c + a = −50 → a = c − 50.
Substitute into (2): 3c + 3(c−50) + 4s = 530 → 6c − 150 + 4s = 530 → 4s = 680 − 6c → s = (340 − 3c)/2.
Substitute into (1): 6c + 7(c−50) + 2s = 990 → 13c − 350 + 2s = 990 → 2s = 1340 − 13c → s = (1340 − 13c)/2.
Equate the two expressions for s: 1340 − 13c = 340 − 3c → 1000 = 10c → c = 100. Then a = 50 and s = 20.
Answer: 100 children, 50 adults, and 20 senior citizens.
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