Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Kelsey 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 $17$ pounds. Large ones, on the other hand, weigh $4$ pounds apiece, and are shipped in a container that weighs $11$ 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? $\newline$ The shipping weight of a full container of either size will be $\_$ pounds if it holds $\_$ helicopters.

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Solve a system of equations using substitution: word problems

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Q. Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

\newline

Kelsey 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

17

pounds. Large ones, on the other hand, weigh

4

pounds apiece, and are shipped in a container that weighs

11

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?

\newline

The shipping weight of a full container of either size will be

\_

pounds if it holds

\_

helicopters.

Define Variables: Let's define variables for the number of helicopters in each container. Let $x$ be the number of small helicopters in a container, and $y$ be the number of large helicopters in a container. We can write two equations based on the given information: $\newline$ For small helicopters: Total weight = weight of container + (weight of one helicopter * number of helicopters) $\newline$ For large helicopters: Total weight = weight of container + (weight of one helicopter * number of helicopters) $\newline$ So we have: $\newline$ $3x + 17 = 4y + 11$ $\newline$ We need to express one variable in terms of the other to use substitution.
Solve for x: Let's solve the first equation for x: $\newline$ $3x + 17 = 4y + 11$ $\newline$ $3x = 4y + 11 - 17$ $\newline$ $3x = 4y - 6$ $\newline$ Now, divide both sides by $3$ to get $x$ in terms of $y$ : $\newline$ $x = \frac{4y - 6}{3}$
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 $3x + 17$ must equal the total weight of a container with large helicopters $4y + 11$ . We can set these two expressions equal to each other: $\newline$ $3x + 17 = 4y + 11$ $\newline$ Substitute the expression for $x$ we found earlier: $\newline$ $3\left(\frac{4y - 6}{3}\right) + 17 = 4y + 11$
Simplify Equation: Simplify the equation: $\newline$ $4y - 6 + 17 = 4y + 11$ $\newline$ $4y + 11 = 4y + 11$ $\newline$ We see that the equation simplifies to a true statement, which means that any value of $y$ 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.