Problem 58 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. You sold two types of scarves at a farmers' market and would like to know which one was more popular. The total number of scarves sold was \(56,\) the yellow scarf cost \(\$ 10,\) and the purple scarf cost \(\$ 11\). If you had total revenue of \(\$ 583,\) how many yellow scarves and how many purple scarves were sold?

Step-by-Step Solution

Answer

33 yellow scarves and 23 purple scarves were sold.

Step 1: Define Variables

Let's define two variables to represent the number of scarves sold. Let \( x \) be the number of yellow scarves sold and \( y \) be the number of purple scarves sold.

Step 2: Formulate Equations

Create a system of equations based on the given conditions. From the problem, we know:1. The total number of scarves sold is 56. This can be expressed as: \[ x + y = 56 \]2. The total revenue from selling the scarves is \\(583, with yellow scarves costing \\)10 each and purple scarves costing \$11 each. This can be expressed as: \[ 10x + 11y = 583 \]

Step 3: Set Up Determinants for Cramer's Rule

Cramer's rule will use determinants of matrices. First, set up the coefficient matrix:\[ A = \begin{bmatrix} 1 & 1 \ 10 & 11 \end{bmatrix} \]Calculate the determinant of \( A \):\[ \Delta = \det(A) = (1)(11) - (1)(10) = 1 \]

Step 4: Create Matrices for Numerators in Cramer's Rule

Substitute the respective columns with the constants to find \( x \) and \( y \):1. For \( x \), replace the first column with the constants: \[ A_x = \begin{bmatrix} 56 & 1 \ 583 & 11 \end{bmatrix} \] The determinant is: \[ \Delta_x = \det(A_x) = (56)(11) - (1)(583) = 616 - 583 = 33 \]

Step 5: Calculate Second Determinant

For \( y \), replace the second column with the constants:\[ A_y = \begin{bmatrix} 1 & 56 \ 10 & 583 \end{bmatrix} \]The determinant is:\[ \Delta_y = \det(A_y) = (1)(583) - (56)(10) = 583 - 560 = 23 \]

Step 6: Solve for Variables Using Cramer's Rule

Use Cramer's Rule to solve for \( x \) and \( y \):1. \( x = \frac{\Delta_x}{\Delta} = \frac{33}{1} = 33 \)2. \( y = \frac{\Delta_y}{\Delta} = \frac{23}{1} = 23 \)Thus, 33 yellow scarves and 23 purple scarves were sold.

Key Concepts

System of Linear Equations

Imagine you have two types of scarves. You know how many were sold in total and how much money you made. But you're not sure how many of each type sold. This is where a system of linear equations comes in handy. These equations use variables to represent unknown values, such as the number of each scarf sold. In our case, we use two equations: one for the total number of scarves and another for the total revenue collected.
\[ x + y = 56 \]
This equation represents the sum of yellow and purple scarves sold, amounting to 56. The other equation is derived from the scarf prices:

• Yellow scarves cost \(10 each.
• Purple scarves cost \)11 each.

With a total of $583 in revenue, our second equation becomes:
\[ 10x + 11y = 583 \]
These two equations together form what is called a system of linear equations, where "linear" means each equation forms a straight line when graphed.

Determinants

Determinants are essential in solving systems of linear equations using Cramer's Rule. They are a special number calculated from a matrix. In simple terms, it's like a "signature" of the matrix that can tell us if the system has a unique solution.
For our problem, the coefficient matrix \( A \) is:
\[ A = \begin{bmatrix} 1 & 1 \ 10 & 11 \end{bmatrix} \]
To find the determinant \( \Delta \) of matrix \( A \), you calculate it as:

• Multiply the diagonal from the top left to the bottom right: \(1 \times 11 \).
• Subtract the product of the diagonal from the top right to the bottom left: \(1 \times 10 \).

The determinant is:
\[ \Delta = (1)(11) - (1)(10) = 1 \]
A non-zero determinant, like 1, tells us that the system of equations has a unique solution.

Matrix Algebra

Matrix algebra helps us organize and manipulate data in a structured way, especially useful with systems of equations. In our example, we used a matrix to represent the coefficients in the equations for the scarves.
Matrices are rectangular arrays of numbers or functions. The coefficient matrix \( A \) shows the factors of \( x \) and \( y \) from our equations. Each row represents an equation, and each column represents a variable. When using Cramer's Rule, matrices become essential:

• The determinant of the matrix helps determine if there is a unique solution.
• If the determinant is zero, we can't use Cramer's Rule directly as it suggests no unique solution.

In our exercise, we also created matrices \( A_x \) and \( A_y \) by substituting constants into the original matrix. This allows us to find the values of \( x \) and \( y \).

Solving Linear Systems

Solving linear systems can be effectively handled using methods like Cramer's Rule, which is suited for systems with the same number of equations as variables. Here, Cramer's Rule uses determinants to find solutions without needing to graph or simplify manually.
First, we calculate the determinant \( \Delta \) of the coefficient matrix \( A \). Then, we set up adjusted matrices, \( A_x \) and \( A_y \), each replacing one column with the constant values from the equations.
By calculating determinants for \( A_x \) and \( A_y \), we apply Cramer's Rule:

• Find \( x = \frac{\Delta_x}{\Delta} \).
• Find \( y = \frac{\Delta_y}{\Delta} \).

This formula gives us exact answers. In this case, 33 yellow scarves \(( x )\) and 23 purple scarves \(( y )\) were sold. Cramer's Rule simplifies the solving process, especially for small systems, by providing clear, numerical results.