Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Dean works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh 3 pounds each, and are shipped in a container that weighs 20 pounds. Large ones, on the other hand, weigh 7 pounds apiece, and are shipped in a container that weighs 16 pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. How many helicopters would fit in either container? What would the total weight be? If either container holds _____ helicopters, it will weigh a total of _____ pounds once it is packed for shipping.

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.  Dean works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh 3 pounds each, and are shipped in a container that weighs 20 pounds. Large ones, on the other hand, weigh 7 pounds apiece, and are shipped in a container that weighs 16 pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. How many helicopters would fit in either container? What would the total weight be?   If either container holds _____ helicopters, it will weigh a total of _____ pounds once it is packed for shipping.

Bytelearn · Accepted Answer

Formulate Equations: Let's denote the number of small helicopters that fit in a container as x and the number of large helicopters that fit in a container as y. We can write two equations to represent the total weight of each packed container. The total weight of a container packed with small helicopters is the weight of the container plus the weight of the helicopters. Similarly, the total weight of a container packed with large helicopters is the weight of the container plus the weight of the helicopters. The equations are as follows:  For small helicopters: 20 + 3x = total weight For large helicopters: 16 + 7y = total weight  Since the problem states that all packed containers will have the same shipping weight, we can set the two expressions equal to each other:  20 + 3x = 16 + 7ySolve Using Substitution: Now we need to solve this system of equations using substitution. First, let's isolate x in the first equation:  3x = 16 + 7y - 20 3x = 7y - 4  Now we can express x in terms of y:  x = (7y - 4) / 3Find Suitable Value of y: Next, we need to find a value of y that makes x an integer since we cannot have a fraction of a helicopter. To do this, we need to find a value of y such that 7y - 4 is divisible by 3. We can start by testing values of y starting from 1 and increasing until we find a suitable value.  Let's test y = 1: 7(1) - 4 = 3, which is divisible by 3, so y = 1 is a solution.  Now we can find the corresponding value of x: x = (7(1) - 4) / 3 x = 3 / 3 x = 1  So, if y = 1, then x = 1.Calculate Total Weight: We have found that both containers can hold 1 helicopter each to have the same shipping weight. Now we need to calculate the total weight for each container when they hold 1 helicopter.  For small helicopters: Total weight = 20 + 3(1) = 20 + 3 = 23 pounds  For large helicopters: Total weight = 16 + 7(1) = 16 + 7 = 23 pounds  Both containers weigh 23 pounds when packed with 1 helicopter.

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