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 farmer looks out his window at his chickens and pigs. He tells his daughter that he sees 62 heads and 190 legs. How many chickens and pigs does the farmer have?

Short answer

The farmer has 29 chickens and 33 pigs.

Step-by-step solution

Define Variables

Let \( x \) represent the number of chickens and \( y \) represent the number of pigs.

Formulate System of Equations

Using the information from the problem, we form the following equations: \[ x + y = 62 \] (Each animal has one head) and \[ 2x + 4y = 190 \] (Chickens have 2 legs each and pigs have 4 legs each).

Set Up Augmented Matrix

Write the system of equations as an augmented matrix: \[\begin{bmatrix}1 & 1 & | & 62 \2 & 4 & | & 190\end{bmatrix}\]

Row Reduction to Echelon Form

Perform row operations to simplify the augmented matrix to row-echelon form. Start by using row 1 to eliminate the first term in row 2: Subtract 2 times row 1 from row 2:\[\begin{bmatrix}1 & 1 & | & 62 \0 & 2 & | & 66\end{bmatrix}\]

Put Matrix in Reduced Row Echelon Form

Divide row 2 by 2 to simplify the entire matrix:\[\begin{bmatrix}1 & 1 & | & 62 \0 & 1 & | & 33\end{bmatrix}\]Now subtract row 2 from row 1 to isolate \( x \):\[\begin{bmatrix}1 & 0 & | & 29 \0 & 1 & | & 33\end{bmatrix}\]

Solve and Interpret the System

From the reduced row-echelon form, we have the equations:\[ x = 29 \] (representing 29 chickens)\[ y = 33 \] (representing 33 pigs).

Key concepts

Augmented Matrix

An augmented matrix is a helpful tool when solving a system of linear equations. It is a matrix that contains both the coefficients of the variables and the constants from the equations themselves. By organizing the system into a matrix form, we simplify the process of finding solutions using matrix operations.

• The left side of the matrix represents the coefficients of each variable in the system.
• The right side (the augmented part, separated by a line) contains the constant terms.

For example, if you have a system of two equations:

• \ \( a_1x + b_1y = c_1 \ \)
• \ \( a_2x + b_2y = c_2 \ \)

The augmented matrix will appear as:\[\begin{bmatrix} a_1 & b_1 & | & c_1 \ a_2 & b_2 & | & c_2 \end{bmatrix}\]By manipulating this matrix through row operations, we aim to find a simpler equivalent system that is easier to solve.

Row Echelon Form

Row echelon form (REF) is an intermediate step toward solving a system of equations using matrices. In this form, the matrix is simplified through row operations into a triangular format.
Here are the key features of a matrix in row echelon form:

• All zero rows are at the bottom of the matrix.
• The leading entry of each non-zero row, called a pivot, is 1 and is to the right of the pivot in the row above.
• Each column containing a pivot has zeros below the pivot.

Taking these steps helps in systematically simplifying the matrix so that further transformations to the reduced row echelon form become straightforward. The process involves row operations such as swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows from each other.

Reduced Row Echelon Form

Reduced row echelon form (RREF) is the simplest form of a matrix that represents a system of linear equations. Once a matrix is in RREF, finding the solution to the system is straightforward.
To be in RREF, a matrix must satisfy certain conditions:

• It is in row echelon form.
• Every pivot (leading 1) is the only non-zero entry in its column.

Through a series of row operations, you transform the augmented matrix until these conditions are met. Once achieved, each variable can be read directly from the matrix, providing a clear and simple solution. For instance, if the RREF of a matrix gives us:\[\begin{bmatrix} 1 & 0 & | & a \ 0 & 1 & | & b \end{bmatrix}\]This corresponds directly to the solutions \( x = a \) and \( y = b \). The clarity offered by the RREF eliminates ambiguity in the solution process.

Variable Definition

Defining variables is a crucial first step in framing any problem within mathematics, especially when dealing with systems of linear equations. It establishes what each symbol represents within the context of the problem.
In our context, variables serve as placeholders for the unknown quantities we're trying to find.

• To start, identify the different unknowns in your problem. For example, the number of chickens and pigs.
• Assign a variable (e.g., \( x \) and \( y \)) to represent each unknown.

This step allows us to translate a real-world situation into a structured mathematical form. In this way, the problem becomes more accessible to mathematical tools and methods, like solving linear equations. When defining variables, clarity is key, as it sets the stage for accurate problem-solving and interpretation. In our exercise, \( x \) is defined as the number of chickens, while \( y \) is the number of pigs.