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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineKelsey works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh 33 pounds each, and are shipped in a container that weighs 1717 pounds. Large ones, on the other hand, weigh 44 pounds apiece, and are shipped in a container that weighs 1111 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? How many helicopters would fit in either container?\newlineThe shipping weight of a full container of either size will be _\_ pounds if it holds _\_ helicopters.

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.\newlineKelsey works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh 33 pounds each, and are shipped in a container that weighs 1717 pounds. Large ones, on the other hand, weigh 44 pounds apiece, and are shipped in a container that weighs 1111 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? How many helicopters would fit in either container?\newlineThe shipping weight of a full container of either size will be _\_ pounds if it holds _\_ helicopters.
  1. Define Variables: Let's define variables for the number of helicopters in each container. Let xx be the number of small helicopters in a container, and yy be the number of large helicopters in a container. We can write two equations based on the given information:\newlineFor small helicopters: Total weight = weight of container + (weight of one helicopter * number of helicopters)\newlineFor large helicopters: Total weight = weight of container + (weight of one helicopter * number of helicopters)\newlineSo we have:\newline3x+17=4y+113x + 17 = 4y + 11\newlineWe need to express one variable in terms of the other to use substitution.
  2. Solve for x: Let's solve the first equation for x:\newline3x+17=4y+113x + 17 = 4y + 11\newline3x=4y+11−173x = 4y + 11 - 17\newline3x=4y−63x = 4y - 6\newlineNow, divide both sides by 33 to get xx in terms of yy:\newlinex=4y−63x = \frac{4y - 6}{3}
  3. Set Equal Weight Equations: Now we can use the fact that the total weight of the containers must be the same when they are full. This means that the total weight of a container with small helicopters 3x+173x + 17 must equal the total weight of a container with large helicopters 4y+114y + 11. We can set these two expressions equal to each other:\newline3x+17=4y+113x + 17 = 4y + 11\newlineSubstitute the expression for xx we found earlier:\newline3(4y−63)+17=4y+113\left(\frac{4y - 6}{3}\right) + 17 = 4y + 11
  4. Simplify Equation: Simplify the equation:\newline4y−6+17=4y+114y - 6 + 17 = 4y + 11\newline4y+11=4y+114y + 11 = 4y + 11\newlineWe see that the equation simplifies to a true statement, which means that any value of yy will satisfy the equation. This suggests that there might be an error in our setup or that additional information is needed to find a unique solution.

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