Select the correct answer.
Determine which situation could be represented by the system of linear equations given below.
[tex]\begin{aligned}
2 x+4 y & =160 \\
x+y & =50
\end{aligned}[/tex]
A. An ice cream shop sells sundaes for $2 each and cones for $4 each. They made $50 over the course of 2 hours and sold a total of 160 items.
B. Jane sells beaded bracelets for $2 each and beaded necklaces for $4 each. She sold 50 pieces of jewelry at the craft fair last weekend and made $160.
C. Kevin bought 2 pairs of jeans and 4 shirts at the mall. He spent $50. Another person bought 1 shirt and 1 pair of jeans for $160.
D. Two principals and four teachers donated a total of 160 cans of soup to the school can drive. The two biggest donors gave 50 cans between them.
Answer (2)
The correct situation represented by the system of equations is Situation B, where Jane sells beaded bracelets for $2 each and necklaces for $4 each, totaling 50 pieces and making $160. This matches the given equations perfectly. Therefore, the answer is Situation B.
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Analyze each situation and create a system of equations.
Compare the created system with the given system.
Situation 1's system is 2 x + 4 y = 50 and x + y = 160 , which doesn't match.
Situation 2's system is 2 x + 4 y = 160 and x + y = 50 , which matches.
Situation 3's system is 2 x + 4 y = 50 and x + y = 160 , which doesn't match.
Situation 4's system is 2 x + 4 y = 160 and x + y = 50 , which matches, but is less accurate than Situation 2.
The correct answer is Situation 2: Jane sells beaded bracelets for $2 each and beaded necklaces for $4 each. She sold 50 pieces of jewelry at the craft fair last weekend and made $160.
Explanation
Analyze the problem and the given equations We are given a system of two linear equations:
2 x + 4 y = 160 x + y = 50
We need to determine which of the four given situations can be represented by this system of equations. We will analyze each situation and create a system of equations for it. Then we will compare the created system with the given system to find the matching one.
Analyze Situation 1 Situation 1: An ice cream shop sells sundaes for $2 each and cones for $4 each. They made $50 over the course of 2 hours and sold a total of 160 items. Let x be the number of sundaes and y be the number of cones. The system of equations is:
2 x + 4 y = 50 x + y = 160
This system does not match the given system.
Analyze Situation 2 Situation 2: Jane sells beaded bracelets for $2 each and beaded necklaces for $4 each. She sold 50 pieces of jewelry at the craft fair last weekend and made $160. Let x be the number of bracelets and y be the number of necklaces. The system of equations is:
2 x + 4 y = 160 x + y = 50
This system matches the given system.
Analyze Situation 3 Situation 3: Kevin bought 2 pairs of jeans and 4 shirts at the mall. He spent $50. Another person bought 1 shirt and 1 pair of jeans for $160. Let x be the price of jeans and y be the price of shirts. The system of equations is:
2 x + 4 y = 50 x + y = 160
This system does not match the given system.
Analyze Situation 4 Situation 4: Two principals and four teachers donated a total of 160 cans of soup to the school can drive. The two biggest donors gave 50 cans between them. Let x be the number of cans donated by principals and y be the number of cans donated by teachers. The system of equations is:
2 x + 4 y = 160 x + y = 50
This system matches the given system.
Determine the correct answer Both Situation 2 and Situation 4 match the given system of equations. However, the question asks to select the correct answer, implying there is only one correct answer. Looking closely at Situation 4, the second sentence says 'The two biggest donors gave 50 cans between them'. This is vague and doesn't necessarily mean that the two principals are the biggest donors. Therefore, Situation 2 is the most accurate representation of the system of equations.
Therefore, the correct answer is Situation 2.
Examples
Systems of linear equations are used in various real-life scenarios, such as determining the optimal mix of products to maximize profit, calculating the nutritional content of a diet, or balancing chemical equations. For example, a farmer might use a system of equations to determine the optimal mix of fertilizers to use on their crops, given the nutrient requirements of the crops and the nutrient content of the fertilizers. Similarly, a chef might use a system of equations to determine the quantities of different ingredients needed to create a dish with a specific nutritional profile. In this case, we used a system of equations to model a sales scenario at a craft fair.
