Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Trevor works in the shipping department of a toy manufacturer. Toy cars weigh $4\,\text{kg}$ apiece and are shipped in a container that weighs $2\,\text{kg}$ when empty. Toy trucks, which weigh $3\,\text{kg}$ apiece, are shipped in a container weighing $9\,\text{kg}$ . When packed with toys and ready for shipment, both kinds of containers have the same number of toys and the same weight. What is the weight of each container? What is the number of toys? $\newline$ Each container weighs $\_\_\_\_$ kilograms and contains $\_\_\_\_$ toys.

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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

Trevor works in the shipping department of a toy manufacturer. Toy cars weigh

4\,\text{kg}

apiece and are shipped in a container that weighs

2\,\text{kg}

when empty. Toy trucks, which weigh

3\,\text{kg}

apiece, are shipped in a container weighing

9\,\text{kg}

. When packed with toys and ready for shipment, both kinds of containers have the same number of toys and the same weight. What is the weight of each container? What is the number of toys?

\newline

Each container weighs

\_\_\_\_

kilograms and contains

\_\_\_\_

toys.

Define Variables: Let's define the variables: $\newline$ Let $x$ be the number of toy cars in the container. $\newline$ Let $y$ be the number of toy trucks in the container. $\newline$ We are given that toy cars weigh $4$ kilograms each and the container weighs $2$ kilograms when empty. Therefore, the total weight of a container filled with toy cars is $4x + 2$ . $\newline$ Similarly, toy trucks weigh $3$ kilograms each and the container weighs $9$ kilograms when empty. Therefore, the total weight of a container filled with toy trucks is $3y + 9$ . $\newline$ We are also given that both kinds of containers have the same number of toys and the same weight. This gives us two equations: $\newline$ $1$ ) $x = y$ (same number of toys) $\newline$ $2$ ) $4x + 2 = 3y + 9$ (same weight)
Substitution to Solve: Now we will use substitution to solve the system of equations. Since $x = y$ , we can substitute $y$ for $x$ in the second equation: $\newline$ $4x + 2 = 3x + 9$
Solve for x: Next, we solve for x: $\newline$ $4x + 2 = 3x + 9$ $\newline$ Subtract $3x$ from both sides: $\newline$ $4x - 3x + 2 = 3x - 3x + 9$ $\newline$ $x + 2 = 9$ $\newline$ Subtract $2$ from both sides: $\newline$ $x = 9 - 2$ $\newline$ $x = 7$
Find Number of Toys: Since $x = y$ , we also have: $\newline$ $y = 7$ $\newline$ Now we know the number of toys in each container is $7$ .
Find Weight of Containers: We can now find the weight of each container. For the container with toy cars: $\newline$ Weight = $4x + 2$ $\newline$ Weight = $4(7) + 2$ $\newline$ Weight = $28 + 2$ $\newline$ Weight = $30$ kilograms
Confirm Weight and Toys: For the container with toy trucks, the weight should be the same, but let's calculate to confirm: $\newline$ Weight = $3y + 9$ $\newline$ Weight = $3(7) + 9$ $\newline$ Weight = $21 + 9$ $\newline$ Weight = $30$ kilograms
Confirm Weight and Toys: For the container with toy trucks, the weight should be the same, but let's calculate to confirm: $\newline$ Weight = $3y + 9$ $\newline$ Weight = $3(7) + 9$ $\newline$ Weight = $21 + 9$ $\newline$ Weight = $30$ kilogramsWe have confirmed that both containers weigh the same, which is $30$ kilograms, and contain the same number of toys, which is $7$ .