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 three exhibits. What are the numbers of children, adults, and seniors citizens that attended these three exhibits?
[tex]\left\{\begin{array}{l}
6 c+7 a+2 s=990 \\
3 c+3 a+4 s=530 \\
2 c+4 a+4 s=480
\end{array}\right.[/tex]
A. 120 children, 30 adults, and 30 senior citizens
B. 20 children, 40 adults, and 70 senior citizens
C. 100 children, 50 adults, and 5 senior citizens
D. 100 children, 50 adults, and 20 senior citizens
Answer (2)
Use elimination to reduce the system to two equations with two unknowns.
Solve for 'a' in terms of 'c' using one of the simplified equations.
Substitute the expression for 'c' into another equation to solve for 'a'.
Substitute the values of 'a' and 'c' into one of the original equations to solve for 's'.
The solution is 100 c hi l d re n , 50 a d u lt s , an d 20 se ni orc i t i ze n s .
Explanation
Understanding the Problem We are given a system of three linear equations with three unknowns: c, a, and s, representing the number of children, adults, and senior citizens, respectively. Our goal is to find the values of c, a, and s that satisfy all three equations.
Stating the Equations The system of equations is:
6 c + 7 a + 2 s = 990 (1) 3 c + 3 a + 4 s = 530 (2) 2 c + 4 a + 4 s = 480 (3)
Eliminating 's' and Expressing 'c' in terms of 'a' We can simplify the system by eliminating one of the variables. Let's eliminate 's' first. Subtract equation (3) from equation (2):
( 3 c + 3 a + 4 s ) − ( 2 c + 4 a + 4 s ) = 530 − 480 c − a = 50 (4) c = a + 50
Creating a Second Equation without 's' Now, multiply equation (1) by 2 to get:
12 c + 14 a + 4 s = 1980 (5) Subtract equation (2) from equation (5):
( 12 c + 14 a + 4 s ) − ( 3 c + 3 a + 4 s ) = 1980 − 530 9 c + 11 a = 1450 (6)
Solving for 'a' Substitute c = a + 50 from equation (4) into equation (6):
9 ( a + 50 ) + 11 a = 1450 9 a + 450 + 11 a = 1450 20 a = 1000 a = 50
Solving for 'c' Now that we have a = 50 , we can find 'c' using equation (4):
c = a + 50 = 50 + 50 = 100
Solving for 's' Substitute the values of 'a' and 'c' into equation (3) to find 's':
2 c + 4 a + 4 s = 480 2 ( 100 ) + 4 ( 50 ) + 4 s = 480 200 + 200 + 4 s = 480 400 + 4 s = 480 4 s = 80 s = 20
Final Answer So, we have c = 100 , a = 50 , and s = 20 . This means there were 100 children, 50 adults, and 20 senior citizens.
Examples
Understanding systems of linear equations is crucial in various real-world applications. For instance, businesses use them to optimize resource allocation, balancing costs and profits. Imagine a company producing multiple products with limited resources; a system of equations can determine the optimal production quantities for each product to maximize profit. Similarly, in nutrition, diet plans can be formulated using systems of equations to meet specific nutritional requirements while adhering to caloric constraints. These systems help in making informed decisions, ensuring efficiency and balance in resource management.
The solution to the equations shows there are 100 children, 50 adults, and 20 senior citizens who attended the exhibits, corresponding to option D. We solved the system using elimination and substitution to find each variable step by step.
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