Problem 53 - 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. $$\left\\{\begin{array}{c} 5 x-3 y+2 z=2 \\ 2 x+2 y-3 z=3 \\ -x+7 y-8 z=4 \end{array}\right.$$

Step-by-Step Solution

Answer

The solutions cannot be provided directly without knowing the specific operations of the graphing utility's matrix capabilities used. However, the method involves converting the system of equations into matrix form, performing the suitable row operations to get the matrix to reduced row echelon form, and interpreting the results.

Step 1: Writing the Augmented Matrix

First, represent the system of equations as an augmented matrix. The matrix will be a 3x4 matrix where the first three columns represent the coefficients of \(x\), \(y\), and \(z\), and the fourth column represents the constants. The augmented matrix is therefore: \[\left[ \begin{array}{ccc|c} 5 & -3 & 2 & 2 \ 2 & 2 & -3 & 3 \ -1 & 7 & -8 & 4 \end{array} \right]\]

Step 2: Matrix operations

Use your graphing utility's matrix capabilities to perform row operations. The primary goal here is to transform the augmented matrix to its reduced row echelon form using row operations like swapping rows, multiplying rows by a non-zero constant, or adding and subtracting rows.

Step 3: Interpreting the Result

Once your matrix is in its reduced row echelon form, interpret the result back into a system of equations to find the solutions for \(x\), \(y\), and \(z\). Keep in mind the correspondence: the first column is for \(x\), the second for \(y\), and the third for \(z\). The last column represents the constants of the equations.

Key Concepts

Matrix Operations

When it comes to solving systems of linear equations, matrix operations are invaluable tools for streamlining the process. These operations involve manipulating matrices—arrays of numbers that represent the coefficients of variables in linear equations—based on a set of rules. To solve a system of equations using matrix operations, we first need to express the system as an augmented matrix. This includes all the coefficients and the constants in a rectangular array.

Some fundamental matrix operations include:

• Row switching (swapping two rows)
• Row multiplication (multiplying all entries in a row by a non-zero constant)
• Row addition (adding or subtracting one row from another)

By applying these operations, one can simplify an augmented matrix into a form where the solution to the system becomes obvious—typically the reduced row echelon form. It's essential to apply these operations correctly, as errors can lead to incorrect solutions or no solution.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficients of the variables and the constants of the equations into a single matrix. The left side of the augmented matrix contains the coefficients of the variables, and a vertical bar separates these from the final column, which holds the constants from each equation.

Example of an Augmented Matrix

In the given exercise, we converted the system of equations into an augmented matrix as follows:
\[\left[ \begin{array}{ccc|c} 5 & -3 & 2 & 2 \ 2 & 2 & -3 & 3 \ -1 & 7 & -8 & 4 \end{array} \right]\]
This process of converting systems of equations into an augmented matrix is the critical first step before proceeding with matrix operations to solve the system.

Reduced Row Echelon Form

The reduced row echelon form (RREF) is a particular form of a matrix achieved through row operations that satisfies the following conditions: each leading coefficient (the first non-zero number from the left in a row) is 1 and is the only non-zero entry in its column, and for any two different rows, the leading coefficient of the upper row is to the left compared to the leading coefficient of the row below.

Additionally, any rows containing only zeroes are at the bottom of the matrix. For a system of linear equations, achieving RREF is desirable because it provides the simplest form to identify the solutions or to determine if no solution exists. In the example from the exercise, the RREF would reflect whether there is a single unique solution set for the variables, there are infinitely many solutions, or there is no solution at all.

Graphing Utilities

Modern technology provides a powerful tool with graphing utilities that help visualize mathematical concepts and perform calculations that can be tedious by hand. In solving systems of linear equations, graphing utilities can quickly turn an augmented matrix into its reduced row echelon form, making the process of finding the solution much easier. Most graphing calculators and computer algebra systems can handle these operations with commands specifically designed for matrix manipulation.

When you input the augmented matrix into a graphing utility, it can perform the necessary row operations and report back the RREF, which you can then interpret to find the solutions for the variables in the system. Utilizing graphing utilities not only saves time but also minimizes the risk of human error during calculation, ensuring more accurate results.