Problem 89 - Precalculus with Limits

Question

In Exercises 85-90, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. \( \left\\{ \begin{array}{l} x + y + z + w = 0 \\ 2x + 3y + z - 2w = 0 \\ 3x + 5y + z = 0 \\ \end{array} \right. \)

Step-by-Step Solution

Answer

From the final system of equations, it can be deduced directly that \(w = 0\). Substituting \(w = 0\) into the second equation gives \(y = z\). Substituting \(w = 0\) and \(y = z\) into the first equation gives \(x = 0 - z - z - 0 = -2z\). Therefore, the system of equations has infinitely many solutions, and can be parameterized as \(x = -2z\), \(y = z\), \(z = z\), and \(w = 0\), where \(z\) can be any real number.

Step 1: Setup the Augmented Matrix

First, setup the augmented matrix corresponding to the given system of equations. The coefficients of the variables x, y, z, w and the constants form the matrix:\[ \begin{pmatrix} 1 & 1 & 1 & 1 & 0 \ 2 & 3 & 1 & -2 & 0 \ 3 & 5 & 1 & 0 & 0 \end{pmatrix}\]

Step 2: Use the row operations to reduce the augmented matrix

Use a graphing utility to perform row operations to reduce the matrix to row-echelon form or reduced row-echelon form. The operations are: \(R2 = R2 - 2*R1\) and \(R3 = R3 - 3*R1\). The interim matrix looks like:\[ \begin{pmatrix} 1 & 1 & 1 & 1 & 0 \ 0 & 1 & -1 & -4 & 0 \ 0 & 2 & -2 & -3 & 0 \end{pmatrix}\]Next, \(R3 = R3 - 2*R2\), results in the row-reduced form:\[ \begin{pmatrix} 1 & 1 & 1 & 1 & 0 \ 0 & 1 & -1 & -4 & 0 \ 0 & 0 & 0 & 5 & 0 \end{pmatrix}\]

Step 3: Convert the Augmented Matrix to System of Equations

Transform the row-reduced matrix back into a system of equations. The system of equations after this transformation are \(x + y + z + w = 0\), \(y - z - 4w = 0\) and \(5w = 0\).

Key Concepts

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It combines both the coefficient matrix and the constants from the system of equations into a single matrix.

• The left side of the matrix contains the coefficients of the variables.
• The rightmost column represents the constants from the equations.

In the given exercise, the augmented matrix is a handy tool because it simplifies handling multiple equations simultaneously. To construct it:
- Take the coefficients of each variable (e.g., x, y, z, w) from the system of equations.- Align these coefficients in rows.- Append the constant term resulting from those equations as an additional column.
For the system\[ \begin{align*} x + y + z + w &= 0 \2x + 3y + z - 2w &= 0 \3x + 5y + z &= 0 \end{align*} \]The corresponding augmented matrix looks like this:\[\begin{pmatrix}1 & 1 & 1 & 1 & 0 \2 & 3 & 1 & -2 & 0 \3 & 5 & 1 & 0 & 0 \\end{pmatrix}.\]

Row Operations

Row operations are algebraic manipulations performed on the rows of the matrix to transform it into a more manageable form. They help in simplifying matrices to form row-echelon or reduced row-echelon forms. Here's how row operations work:

• Exchange: Swap two rows.
• Scaling: Multiply all entries of a row by a non-zero constant.
• Replacement: Add or subtract a multiple of one row to another to achieve elimination.

In the exercise, you used row operations as follows:- Subtraction of multiples of rows to eliminate variables step by step, which helps clarify the relationships between them.
For example:
- The operation \(R2 = R2 - 2*R1\) transforms the second row, helping in eliminating the \(x\) term in Row 2.
- Similarly, \(R3 = R3 - 3*R1\) modifies Row 3, making it easier to isolate variables.
This iterative process continues until the matrix is in a simpler form, like row-echelon or reduced row-echelon form.

Reduced Row-Echelon Form

The reduced row-echelon form (RREF) is a simplified version of a matrix that makes it easier to identify solutions to a system of equations. It's a matrix form where:

• Leading coefficients (known as pivots) are 1.
• Each pivot is the only non-zero entry in its column.
• Rows with all zero elements are at the bottom of the matrix.

For our exercise, performing operations such as \(R3 = R3 - 2*R2\) leads the matrix into its reduced row-echelon form:\[\begin{pmatrix}1 & 1 & 1 & 1 & 0 \0 & 1 & -1 & -4 & 0 \0 & 0 & 0 & 5 & 0 \\end{pmatrix}\]This form indicates that:

• The system's equations can be easily rewritten by tracing back through the pivots.
• Any variables without pivots in a column are free variables, contributing to multiple solutions or dependencies.

The RREF provides a visual way to solve the system or understand its potential infinite solutions if present.