Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{c} 2 x+5 y+w=11 \\ x+4 y+2 z-2 w=-7 \\ 2 x-2 y+5 z+w=3 \\ x-3 w=-1 \end{array}\right.$$

Step-by-Step Solution

Answer

Use a graphing utility to perform row reductions on the matrix \(\begin{bmatrix} 2 & 5 & 0 & 1 & 11 \ 1 & 4 & 2 & -2 & -7 \ 2 & -2 & 5 & 1 & 3 \ 1 & 0 & 0 & -3 & -1 \end{bmatrix}\). The final reduced form of the matrix will give the solutions for the variables \(x\), \(y\), \(z\), and \(w\) in the system of equations.

Step 1: Represent the system as a matrix

The system can be represented as an augmented matrix. The augmented matrix, \(A|b\), for our given system of equations is \(\begin{bmatrix} 2 & 5 & 0 & 1 & 11 \ 1 & 4 & 2 & -2 & -7 \ 2 & -2 & 5 & 1 & 3 \ 1 & 0 & 0 & -3 & -1 \end{bmatrix}\)

Step 2: Row Reduction

Perform row reduction operations to get the matrix into row-echelon form. Use the capabilities of a graphing utility to do this.

Step 3: Read off solutions

Once the matrix is reduced, the solutions to the system of linear equations can be read off by reading across each row to establish the value of each variable.

Key Concepts

Matrix Representation

In solving linear systems, displaying the equations in matrix form is a crucial step. Matrices help organize the coefficients of variables in a clear structure.
The system of equations can be translated into a matrix consisting only of the coefficients of the variables.
For example, the equation system:

• \(2x + 5y + w = 11\)
• \(x + 4y + 2z - 2w = -7\)
• \(2x - 2y + 5z + w = 3\)
• \(x - 3w = -1\)

is represented in a matrix as:

• The rows tend to be the equations.
• The columns usually stand for each variable, each containing the coefficients from the equations.

This arrangement benefits computational solutions by simplifying operations and setting the stage for further manipulation, helping us eventually solve for each variable.

Augmented Matrix

An augmented matrix is a key concept when solving linear equations. It extends the matrix representation by including the constants from the right side of each equation in an extra column.
This extra column helps pair the corresponding equations with their results:
The augmented matrix takes the shape:\[\begin{bmatrix} 2 & 5 & 0 & 1 & 11 \1 & 4 & 2 & -2 & -7 \2 & -2 & 5 & 1 & 3 \1 & 0 & 0 & -3 & -1 \end{bmatrix}\]

• The first four columns are identical to our equation's coefficients.
• The last column is made up of the constants from each equation.

This augmented matrix format is practical for applying operations to solve the system, as it allows for efficient use of computational resources.

Row-Echelon Form

Once in augmented form, the next task is to simplify the matrix using operations until it reaches row-echelon form (REF).
The REF is characterized by having leading ones in each row and zeros below each leading one:

• Rows are organized such that each leading coefficient of a row is to the right of the leading coefficient of the row above it.
• The entries below a leading 1 are all zeros.

This simplification process can be done manually or using technology, such as a graphing utility. The REF makes it much easier to observe any solutions by highlighting where variables stand in relation to each other.
For this reason, reaching the REF is a crucial step toward solving systems of equations using matrices.

Graphing Utility

A graphing utility can significantly streamline the process of solving linear systems with matrices.
These tools come equipped with functions to perform row operations and transform matrices to row-echelon form effortlessly.
Here's how they help:

• Calculations are faster and more precise, reducing human error.
• They allow visualization of transformations, giving learners a better understanding of matrix manipulations.

By automating these processes, a graphing utility can quickly provide row-echelon or even reduced row-echelon forms. Thus, enabling a swift determination of the solutions without getting bogged down by arithmetic details. Utilizing graphing utilities thus enhances both the efficiency and accuracy in solving complex systems of linear equations.