Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Gavin works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh $1$ pound 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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Algebra 1

Solve a system of equations using any method: word problems

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

\newline

Gavin works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh

1

pound 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 denote the number of small helicopters that fit into a small container as $x$ and the number of large helicopters that fit into a large container as $y$ . We need to find a system of equations that represents the total weight of each full container.
Calculate Small Container Weight: The total weight of a full small container can be represented by the weight of the container plus the weight of the small helicopters it holds. This gives us the equation: $\newline$ $1 \times x + 17 = \text{total weight}$
Calculate Large Container Weight: Similarly, the total weight of a full large container can be represented by the weight of the container plus the weight of the large helicopters it holds. This gives us the equation: $4y + 11 = \text{total weight}$
Set Equations Equal: Since the problem states that all of the packed containers will have the same shipping weight, we can set the two equations equal to each other to find the relationship between $x$ and $y$ : $1 \times x + 17 = 4 \times y + 11$
Isolate Variable $x$ : To solve for one of the variables, we can rearrange the equation to isolate $x$ :
$x = 4y + 11 - 17$
$x = 4y - 6$
Express in Terms of Total Weight: Now we need to find a common weight that both types of containers can have when full. Since we don't have a specific weight to work with, we can choose a variable to represent the total weight. Let's call it $W$ . We can then express $x$ and $y$ in terms of $W$ : $W = 1x + 17$ $W = 4y + 11$
Substitute to Find Total Weight: We can substitute the expression for $x$ from Step $5$ into the first equation to find $W$ in terms of $y$ :
$W = 1(4y - 6) + 17$
$W = 4y - 6 + 17$
$W = 4y + 11$
Set Equal to Find Solution: Now we have two expressions for $W$ , one in terms of $x$ and one in terms of $y$ . Since they are equal, we can set them equal to each other: $\newline$ $1x + 17 = 4y + 11$
Set Equal to Find Solution: Now we have two expressions for $W$ , one in terms of $x$ and one in terms of $y$ . Since they are equal, we can set them equal to each other: $\newline$ $1x + 17 = 4y + 11$ We already have this equation from Step $4$ , so we are not getting new information. We need to find a specific solution for $x$ and $y$ that satisfies the equation. Since the problem does not provide additional constraints, we can choose a value for $y$ that makes $x$ an integer. Let's choose $y = 1$ to see if it gives us a valid solution: $\newline$ $x = 4(1) - 6$ $\newline$ $x$ $0$ $\newline$ $x$ $1$