Question

Find the solution of the given problem by: (a) creating an appropriate system of linear equations (b) forming the augmented matrix that corresponds to this system (c) putting the augmented matrix into reduced row echelon form (d) interpreting the reduced row echelon form of the matrix as a solution. A lady buys 20 trinkets at a yard sale. The cost of each trinket is either $$\$ 0.30$$ or $$\$ 0.65 .$$ If she spends $$\$ 8.80$$, how many of each type of trinket does she buy?

Short answer

12 trinkets at $0.30 and 8 trinkets at $0.65.

Step-by-step solution

Define the variables

Let's denote the number of trinkets costing \(0.30 each as \( x \) and the number of trinkets costing \)0.65 each as \( y \).

Set up the system of equations

We have two constraints: the total number of trinkets is 20, and the total cost is $8.80. This gives us the system of linear equations: 1. \( x + y = 20 \) 2. \( 0.30x + 0.65y = 8.80 \)

Form the augmented matrix

Convert the system of equations into an augmented matrix: \[ \begin{bmatrix} 1 & 1 & | & 20 \ 0.3 & 0.65 & | & 8.80 \end{bmatrix} \]

Perform row operations to obtain row echelon form

To simplify the matrix, our first step is to make the leading coefficient (in the first row) a 1 and eliminate the first element of the second row: Subtract 0.3 times the first row from the second row: \[ \begin{bmatrix} 1 & 1 & | & 20 \ 0 & 0.35 & | & 2.80 \end{bmatrix} \]

Reduce to row echelon form

Now divide the second row by 0.35 to get the leading 1:\[ \begin{bmatrix} 1 & 1 & | & 20 \ 0 & 1 & | & 8 \end{bmatrix} \]

Back-substitute to find solutions

Using the second row \( y = 8 \), substitute back into the first equation to find \( x \):\( x + 8 = 20 \) \( x = 12 \).

Final interpretation

From our back-substitution, the solution implies that the lady bought 12 trinkets at $0.30 each and 8 trinkets at $0.65 each.

Key concepts

Systems of Equations

When faced with a real-world problem, like determining how many of each type of trinket the lady purchased, a system of equations provides a structured way to find solutions. In this context, a system of equations consists of more than one equation that shares common variables. Here, we use two equations:

• One for the total count of items: \( x + y = 20 \)
• Another for the total cost: \( 0.30x + 0.65y = 8.80 \)

Systems of equations are incredibly useful in various fields like engineering, economics, and management to solve complex problems. They help us figure out values that satisfy all equations involved. Understanding how to set them up from a word problem is the first vital step towards using them effectively.

Augmented Matrix

An augmented matrix is a compact and systematic way to represent a system of linear equations. Each row of the matrix corresponds to one equation, and each column corresponds to the coefficients of the variables. The last column contains the constants from each equation's right-hand side.In the given problem, the augmented matrix form is:\[\begin{bmatrix} 1 & 1 & | & 20 \ 0.30 & 0.65 & | & 8.80 \end{bmatrix}\]This matrix layout allows us to apply matrix operations efficiently to solve the system of equations. Augmented matrices streamline the method, providing clarity and simplicity when dealing with more than two equations or unknowns. It's a crucial tool for anyone solving linear algebra problems.

Reduced Row Echelon Form

The reduced row echelon form (RREF) of a matrix is a simplified form that has special properties making it easier to solve equations. Transforming a matrix into RREF involves performing row operations, which include row switching, row multiplication, and row addition. The goal is to make the matrix into a form where the solution to the system of equations is apparent.For the trinket problem, achieving RREF gives us:\[\begin{bmatrix} 1 & 1 & | & 20 \ 0 & 1 & | & 8 \end{bmatrix}\]This form shows that \( y = 8 \) and simplifies the equation \( x + y = 20 \) to easily solve for \( x \). Being familiar with RREF allows students to solve complex systems more easily by revealing the relationships between variables directly from the matrix.

Matrix Operations

Matrix operations are fundamental techniques for manipulating matrices to solve systems of equations. These operations include:

• Row Switching: Moving rows around for easier computation.
• Row Multiplication: Multiplying all elements of a row by a non-zero scalar to change its relative weight.
• Row Addition: Adding or subtracting the elements of one row to/from another to create zeros below leading ones.

In the given problem, these operations helped transform the augmented matrix into a reduced row echelon form where solutions could be easily seen. Understanding these basic operations is essential as they are widely used in computer graphics, data analysis, engineering, and various applications of linear algebra. By mastering these operations, students can solve equations faster and more intuitively.