Problem 57 - College Algebra with Corequisite Support

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

30 straw hats, 40 beanies, and 30 cowboy hats.

Step 1: Define Variables

Let \( s \) be the number of straw hats, \( b \) be the number of beanies, and \( c \) be the number of cowboy hats sold. These will be our variables of interest that we want to solve for.

Step 2: Setup the System of Equations

We have three pieces of information to create the system of equations. 1. **Total Hats**: Altogether, 100 hats were sold: \( s + b + c = 100 \).2. **Total Income**: The total income was $1119: \( 13.99s + 7.99b + 14.49c = 1119 \).3. **Beanie Relationship**: There were 10 more beanies than cowboy hats: \( b = c + 10 \).

Step 3: Transform Equation 3

Substitute the third equation, \( b = c + 10 \), into the other equations (to simplify) where relevant. This reduces the system to two variables \(s, c\) in equations 1 and 2.

Step 4: Substitute into the Equations

Substitute \( b = c + 10 \) into equation 1: - \(s + (c + 10) + c = 100\)- Simplifies to: \( s + 2c = 90 \).

Step 5: Substitute b in the Income Equation

Substitute \( b = c + 10 \) into equation 2:- \( 13.99s + 7.99(c + 10) + 14.49c = 1119 \)- Expanding gives: \( 13.99s + 7.99c + 79.9 + 14.49c = 1119 \)- Which simplifies to: \( 13.99s + 22.48c = 1039.10 \).

Step 6: Matrix Representation

Represent the system of equations \( s + 2c = 90 \) and \( 13.99s + 22.48c = 1039.10 \) as a matrix equation:\[\begin{bmatrix} 1 & 2 \ 13.99 & 22.48 \end{bmatrix}\begin{bmatrix} s \ c \end{bmatrix} = \begin{bmatrix} 90 \ 1039.10 \end{bmatrix}\]

Step 7: Find the Inverse of the Coefficient Matrix

Calculate the inverse of the coefficient matrix: \[A^{-1} = \frac{1}{det(A)} \cdot \begin{bmatrix} A_{22} & -A_{12} \ -A_{21} & A_{11} \end{bmatrix}\]Evaluate the determinant and inverse.

Step 8: Compute the Solution Using the Inverse

Multiply the inverse matrix by the constants to find solutions for \( s \) and \( c \). From \( A^{-1}b = x \), solve to get values of \( s, c \).

Step 9: Substitute Back to Find b

Having found \( s \) and \( c \), use \( b = c + 10 \) to find \( b \).

Step 10: Verify the Solution

Check solutions by substituting back into the original equations to ensure the system is correctly satisfied.

Key Concepts

Matrix Inverse

When dealing with systems of equations, the matrix inverse is a powerful tool that can simplify the process. The idea is that if you have a matrix equation like \( AX = B \), where \( A \) is a square matrix, the solution \( X \) can be acquired by multiplying both sides of the equation by the inverse of \( A \) to get \( X = A^{-1}B \). So, what exactly is a matrix inverse?

• **Definition:** The inverse of a matrix \( A \) is another matrix, denoted \( A^{-1} \), such that when it is multiplied by \( A \), yields the identity matrix: \( AA^{-1} = I \).
• **Existence:** Not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.

In the context of our problem, we used the inverse of the coefficient matrix to find the number of straw hats and cowboy hats sold. Calculating the inverse requires knowing the determinant and then applying a formula involving the elements of the matrix. Once the inverse is found, solving the matrix equation yields the values needed for the solution.

Linear Algebra

Linear algebra is the branch of mathematics dealing with vectors, vector spaces, matrices, and linear transformations. It plays a vital role in solving systems of equations like our clothing store problem.

• **Vectors and Matrices:** A matrix is simply an array of numbers, and each element of the matrix is called a vector. For example, in our problem, we set up a matrix equation to represent the coefficients of the system of equations.
• **Operations:** Linear algebra allows various operations, such as addition, multiplication, and finding inverses, which are critical when solving matrix equations.

In our example, linear algebra helped transform the system of equations of the hat sales into a neat matrix form. This enabled us to use linear algebra techniques like calculating the inverse to effectively find the solution to the problem. By utilizing these techniques, complex systems of equations become much more manageable.

Solving Word Problems

Solving word problems can often be challenging because they require translating real-world situations into mathematical terms. Here's how to approach solving a word problem like the one in our clothing store exercise:

• **Understand the Problem:** Carefully read the problem to understand what is being asked. Identify relevant information, such as total sales or relationships between quantities.
• **Define Variables:** Use variables to represent unknown quantities. In our exercise, \( s \), \( b \), and \( c \) represented straw hats, beanies, and cowboy hats, respectively.
• **Set Up Equations:** Use the information provided to form equations. For example, total sales and pricing details led to two equations in the store's exercise.
• **Solve the Equations:** Choose an appropriate method to solve the system. We used matrix inverse techniques, but other methods could be graphing, substitution, or elimination.
• **Verify the Solution:** Substitute the solutions back into the original problem to ensure they make sense and satisfy all conditions.

By taking the problem apart and following these steps, solving word problems becomes a more structured and less daunting task. This organized approach allows for logical problem-solving and ensures all aspects of the problem are addressed.