Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Scarlett works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh $1$ kilogram each, and are shipped in a container that weighs $7$ kilograms. Large ones, on the other hand, weigh $3$ kilograms apiece, and are shipped in a container that weighs $5$ kilograms. 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? $\newline$ If either container holds _ helicopters, it will weigh a total of _ kilograms once it is packed for shipping.

View step-by-step help

Home

Math Problems

Algebra 1

Solve a system of equations using substitution: word problems

Full solution

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

\newline

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

1

kilogram each, and are shipped in a container that weighs

7

kilograms. Large ones, on the other hand, weigh

3

kilograms apiece, and are shipped in a container that weighs

5

kilograms. 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?

\newline

If either container holds _____ helicopters, it will weigh a total of _____ kilograms once it is packed for shipping.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of small helicopters in a container. $\newline$ Let $y$ be the number of large helicopters in a container. $\newline$ The total weight of a container with small helicopters is $1x + 7$ (since each small helicopter weighs $1$ kilogram and the container itself weighs $7$ kilograms). $\newline$ The total weight of a container with large helicopters is $3y + 5$ (since each large helicopter weighs $3$ kilograms and the container itself weighs $5$ kilograms). $\newline$ We are given that the total weight of both types of containers is the same. Therefore, we can write the following equation: $\newline$ $1x + 7 = 3y + 5$
Solve for x: Now, let's solve for one of the variables. We can solve for $x$ in terms of $y$ or $y$ in terms of $x$ . Let's solve for $x$ :
$1x + 7 = 3y + 5$
$1x = 3y + 5 - 7$
$1x = 3y - 2$
Now we have an equation for $x$ in terms of $y$ .
Set Up Equations: Since we want to find the number of helicopters that would fit in either container, we need to set up a second equation that represents the condition that the containers will have the same shipping weight. We already have the equation for the total weight of each container, so we can use the equation we just found $1x = 3y - 2$ to express $x$ in terms of $y$ and substitute it into the first equation to find the value of $y$ .
Substitute Expression: Substitute the expression for $x$ into the first equation: $\newline$ $1x + 7 = 3y + 5$ $\newline$ $(3y - 2) + 7 = 3y + 5$ $\newline$ $3y + 5 = 3y + 5$ $\newline$ This simplifies to $0 = 0$ , which is always true, indicating that we have an infinite number of solutions. This means that for any number of large helicopters $y$ , there is a corresponding number of small helicopters $x$ that will make the containers weigh the same. However, we need to find the specific number of helicopters that would fit in either container, so we need to re-evaluate our approach.
Correct Approach: We realize that we made a mistake in our approach. We should have set up two separate equations, one for the total weight of the small helicopter container and one for the total weight of the large helicopter container, and then set them equal to each other. Let's correct this: $\newline$ For small helicopters: Total weight = $1x + 7$ $\newline$ For large helicopters: Total weight = $3y + 5$ $\newline$ Since the total weights are equal, we set the two equations equal to each other: $\newline$ $1x + 7 = 3y + 5$