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Martin works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh 22 pounds each, and are shipped in a container that weighs 2020 pounds. Large ones, on the other hand, weigh 44 pounds apiece, and are shipped in a container that weighs 1010 pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. What would the total weight be?\newlineWrite a system of equations, graph them, and type the solution.\newline___ pounds\newline

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Q. Martin works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh 22 pounds each, and are shipped in a container that weighs 2020 pounds. Large ones, on the other hand, weigh 44 pounds apiece, and are shipped in a container that weighs 1010 pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. What would the total weight be?\newlineWrite a system of equations, graph them, and type the solution.\newline___ pounds\newline
  1. Define variables: Let's define the variables: Let xx be the number of small helicopters in each container, and yy be the number of large helicopters in each container.
  2. Equation for small helicopters: Write the equation for the total weight of a container with small helicopters: Total weight = weight of container + (number of helicopters ×\times weight per helicopter). So, for small helicopters, the equation is: 20+2x=W20 + 2x = W, where WW is the total weight of the packed container.
  3. Equation for large helicopters: Write the equation for the total weight of a container with large helicopters: Similarly, for large helicopters, the equation is: 10+4y=W10 + 4y = W.
  4. Relationship between xx and yy: Set the equations equal to each other to find a relationship between xx and yy: 20+2x=10+4y20 + 2x = 10 + 4y.
  5. Solve for one variable: Simplify the equation to solve for one variable: 2x−4y=−102x - 4y = -10. Divide everything by 22 to simplify: x−2y=−5x - 2y = -5.
  6. Assume equal total weight: We need another equation to solve this system. Let's assume the total weight of each packed container is the same, so we set WW equal in both equations: 20+2x=10+4y20 + 2x = 10 + 4y.
  7. Solve for x in terms of y: Solve for x in terms of y from the simplified equation: x=2y−5x = 2y - 5.
  8. Substitute xx to find WW: Substitute xx in one of the original weight equations to find WW: 20+2(2y−5)=10+4y20 + 2(2y - 5) = 10 + 4y. Simplify to find WW: 20+4y−10=10+4y20 + 4y - 10 = 10 + 4y.

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