Problem 57 - College Algebra

Question

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A clothing store needs to order new inventory. It has three different types of hats for sale: straw hats, beanies, and cowboy hats. The straw hat is priced at \(\$ 13.99\) , the beanie at \(\$ 7.99,\) and the cowboy hat at \(\$ 14.49\) . If 100 hats were sold this past quarter, \(\$ 1,119\) was taken in by sales, and the amount of beanies sold was 10 more than cowboy hats, how many of each should the clothing store order to replace those already sold?

Step-by-Step Solution

Answer

Order 50 straw hats, 40 beanies, and 10 cowboy hats.

Step 1: Identify Variables

Let's denote the number of straw hats as \(s\), the number of beanies as \(b\), and the number of cowboy hats as \(c\).

Step 2: Write System of Equations

From the problem, we have the following equations: \(s + b + c = 100\) for the total number of hats sold, \(13.99s + 7.99b + 14.49c = 1119\) for the total revenue, and \(b = c + 10\) because the beanies sold were 10 more than cowboy hats.

Step 3: Express as a Matrix Equation

The system of equations can be expressed in matrix form as: \[\begin{bmatrix} 1 & 1 & 1 \13.99 & 7.99 & 14.49 \0 & 1 & -1 \end{bmatrix}\begin{bmatrix}s \b \c\end{bmatrix} =\begin{bmatrix}100 \1119 \10\end{bmatrix}\]

Step 4: Find the Inverse of the Coefficient Matrix

Calculate the inverse of the coefficient matrix \(A\): \[A^{-1} = \begin{bmatrix} 1 & 1 & 1 \13.99 & 7.99 & 14.49 \0 & 1 & -1 \end{bmatrix}^{-1}\]Find \(A^{-1}\) using a calculator or matrix software.

Step 5: Multiply by the Inverse

Use the inverse matrix to solve for \(\begin{bmatrix}s \ b \ c\end{bmatrix}\) by calculating \[\begin{bmatrix}s \ b \ c\end{bmatrix} = A^{-1} \begin{bmatrix}100 \ 1119 \ 10\end{bmatrix}\]

Step 6: Calculate the Result

Perform the matrix multiplication to find the values of \(s\), \(b\), and \(c\). The solution will give the number of straw hats, beanies, and cowboy hats the store should order.

Key Concepts

Matrix Inverse

The concept of the matrix inverse is crucial for solving systems of linear equations. Matrices can serve as a powerful tool in problem-solving within algebra. A matrix inverse acts like the reciprocal of a number but for matrices. For a square matrix \( A \), if there is another matrix \( A^{-1} \) such that \( AA^{-1} = A^{-1}A = I \) (where \( I \) is the identity matrix), then \( A^{-1} \) is called the inverse of \( A \).
To find a matrix inverse, the matrix must be square (same number of rows and columns) and have a non-zero determinant. Methods to find inverses include using row operations or applying formulas that work for 2x2 matrices.
When solved correctly, invertible matrices simplify equations, allowing easier computation of variable values.

Matrix Equation

A matrix equation expresses systems of linear equations in a compact form, using matrices for coefficients and vector matrices for variables and constants. Consider our system of equations concerning the sales of hats. Instead of handling lengthy expressions, we can express all equations as a single matrix equation:

• Coefficient Matrix: \( \begin{bmatrix} 1 & 1 & 1 \ 13.99 & 7.99 & 14.49 \ 0 & 1 & -1 \end{bmatrix} \)
• Variable Matrix: \( \begin{bmatrix} s \ b \ c \end{bmatrix} \)
• Constants Matrix: \( \begin{bmatrix} 100 \ 1119 \ 10 \end{bmatrix} \)

The matrix equation then looks like: \[ A \mathbf{x} = \mathbf{b} \] where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the variable matrix, and \( \mathbf{b} \) is the constants matrix.
Solving this form using matrix algebra, particularly the matrix inverse, offers efficiency and prevents mathematical errors from manual computation.

Linear Equations

Linear equations are the foundation for forming matrix equations. In a linear equation, each term is either a constant or the product of a constant and a single variable. The hallmark of linear equations is that they graph as straight lines or planes when existing in multi-variable contexts.
Our problem involves three linear equations:

• \( s + b + c = 100 \)
• \( 13.99s + 7.99b + 14.49c = 1119 \)
• \( b = c + 10 \)

This set, when written, represents the conditions set by the problem. These equations also highlight relationships between the number of hats and the total sales figures that are central to finding the solution. Converting these into a solitary matrix equation simplifies the overall solving process.

Problem Solving

Approaching problem-solving through systems of equations and matrices requires understanding how each piece of information connects. When given a practical scenario, like ordering hats for a store, the first step is identifying the requirements in terms of conditions or relationships.
Here's how to approach it:

• Define Variables: Assign a variable to each unknown element.
• Write Equations: Translate each statement of the problem into an equation, capturing relations among variables.
• Formulate a Matrix Equation: Organize all equations into a systematic matrix to simplify handling them mathematically.
• Apply Matrix Inversion: If applicable, compute the inverse to uncover the solution effectively.

Translating real-world scenarios into mathematical expressions helps uncover valuable insights, turning complex problems into manageable algebraic forms.

Algebra

Algebra is a branch of mathematics responsible for handling variables and the rules for manipulating them. It gives us the frameworks and tools needed for organizing and analyzing mathematical relations and functions.
Understanding algebra starts with knowing how to work equations, which involves:

• Using Variables: Placeholders like \( s \), \( b \), and \( c \) in expressions.
• Building Expressions: Combinations of variables and constants using operators like addition and multiplication.
• Solving Equations: Determining the value of variables that satisfy a given equation.

By mastering algebraic concepts, students can tackle various types of problems, whether they involve simple one-variable equations or complex systems expressed through matrices. Key to this is familiarity with operations, principles, and interconnections within equations.