Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Dean 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 $20$ pounds. Large ones, on the other hand, weigh $7$ pounds apiece, and are shipped in a container that weighs $16$ pounds. 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 _ pounds once it is packed for shipping.

View step-by-step help

Home

Math Problems

Precalculus

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

Dean 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

20

pounds. Large ones, on the other hand, weigh

7

pounds apiece, and are shipped in a container that weighs

16

pounds. 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 _____ pounds once it is packed for shipping.

Formulate Equations: Let's denote the number of small helicopters that fit in a container as $x$ and the number of large helicopters that fit in a container as $y$ . We can write two equations to represent the total weight of each packed container. The total weight of a container packed with small helicopters is the weight of the container plus the weight of the helicopters. Similarly, the total weight of a container packed with large helicopters is the weight of the container plus the weight of the helicopters. The equations are as follows: $\newline$ For small helicopters: $20 + 3x =$ total weight $\newline$ For large helicopters: $16 + 7y =$ total weight $\newline$ Since the problem states that all packed containers will have the same shipping weight, we can set the two expressions equal to each other: $\newline$ $20 + 3x = 16 + 7y$
Solve Using Substitution: Now we need to solve this system of equations using substitution. First, let's isolate $x$ in the first equation: $\newline$ $3x = 16 + 7y - 20$ $\newline$ $3x = 7y - 4$ $\newline$ Now we can express $x$ in terms of $y$ : $\newline$ $x = \frac{7y - 4}{3}$
Find Suitable Value of $y$ : Next, we need to find a value of $y$ that makes $x$ an integer since we cannot have a fraction of a helicopter. To do this, we need to find a value of $y$ such that $7y - 4$ is divisible by $3$ . We can start by testing values of $y$ starting from $1$ and increasing until we find a suitable value. $\newline$ Let's test $y = 1$ : $\newline$ $7(1) - 4 = 3$ , which is divisible by $3$ , so $y = 1$ is a solution. $\newline$ Now we can find the corresponding value of $x$ : $\newline$ $y$ $3$ $\newline$ $y$ $4$ $\newline$ $y$ $5$ $\newline$ So, if $y = 1$ , then $y$ $5$ .
Calculate Total Weight: We have found that both containers can hold $1$ helicopter each to have the same shipping weight. Now we need to calculate the total weight for each container when they hold $1$ helicopter. $\newline$ For small helicopters: $\newline$ Total weight = $20 + 3(1) = 20 + 3 = 23$ pounds $\newline$ For large helicopters: $\newline$ Total weight = $16 + 7(1) = 16 + 7 = 23$ pounds $\newline$ Both containers weigh $23$ pounds when packed with $1$ helicopter.