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APPLICATION Zoe must ship 532 tubas and 284 kettledrums from her warehouse to a store across the country. A truck rental company offers two sizes of trucks. A small truck will hold 5 tubas and 7 kettledrums. A large truck will hold 12 tubas and 4 kettledrums. If she wants to fill each truck so that the cargo won't shift, how many small and large trucks should she rent? a. Define variables and write a system of equations to find the number of small trucks and the number of large trucks Zoe needs to ship the instruments. (Hint: Write one equation for each instrument.) (Ti ) b. Write a matrix that represents the system. (a) c. Perform row operations to transform the matrix into a solution matrix. d. Write a sentence describing the real-world meaning of the solution.

Step by step solution

01

Define Variables

Let \( x \) represent the number of small trucks, and \( y \) represent the number of large trucks. These variables help us model the scenario mathematically.

02

Set Up System of Equations

Using the problem description:1. For tubas: \( 5x + 12y = 532 \).2. For kettledrums: \( 7x + 4y = 284 \).These equations capture the packing capacity of both truck types for each instrument.

03

Write the Matrix

Translate the system of equations into an augmented matrix:\[\begin{bmatrix}5 & 12 & | & 532 \7 & 4 & | & 284\end{bmatrix}\]

04

Apply Row Operations

Perform row operations to reduce to row-echelon form:1. Multiply the first row by 7 and the second by 5:\[\begin{bmatrix}35 & 84 & | & 3724 \35 & 20 & | & 1420\end{bmatrix}\]2. Subtract the second row from the first:\[\begin{bmatrix}0 & 64 & | & 2304 \35 & 20 & | & 1420\end{bmatrix}\]3. Solve for \( y \) in the second row:\( y = \frac{2304}{64} = 36 \).4. Substitute \( y = 36 \) back into the first row to solve for \( x \): \( 5x + 12(36) = 532 \) which simplifies to \( 5x = 100 \), thus \( x = 20 \).

05

Interpret the Solution

The solution \( x = 20 \) and \( y = 36 \) means Zoe needs 20 small trucks and 36 large trucks to transport the tubas and kettledrums.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Representation

When solving systems of equations, one powerful method is using matrices. Think of a matrix as a way to organize multiple equations neatly. In our scenario with Zoe, we have two equations to handle: one for tubas and another for kettledrums.
Converting these equations into a matrix gives us a clear structure to work with. Each row in the matrix corresponds to one of the equations. By writing the equations in matrix form, we streamline our calculations and make it easier to apply row operations.
This method is similar to converting words into a mathematical language that lays a foundation for numerical solutions.

Real-World Applications

Systems of equations are not just theoretical; they have practical applications in real-life scenarios. Zoe’s problem of transporting musical instruments using available truck capacities is a perfect example.
By defining variables and establishing a system of equations, Zoe can accurately calculate how many trucks she needs. This helps her choose the most efficient and cost-effective method to ship her items.

• Transportation logistics often involve such planning
• They ensure maximum space utilization
• This method avoids unnecessary expenses

Understanding the system of equations can be crucial in planning and executing logistical operations effectively.

Row Operations

Row operations are a set of techniques used to simplify matrices. They help us find solutions to systems of equations like the one Zoe is working on. By performing operations directly on the rows of the matrix, we can systematically reduce it to find our answers.
In Zoe's case, we used these operations to transform the matrix into a form where solving becomes straightforward. Here are common row operations you might use:

• Multiplying a row by a scalar
• Switching two rows
• Adding or subtracting a multiple of one row from another

These operations allow us to isolate variables efficiently. By following the steps methodically, Zoe was able to find she needed 20 small trucks and 36 large ones. This strategy is particularly useful for solving larger systems of equations quickly.