Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system by Gaussian elimination. $$ \begin{array}{r} x+y=2 \\ x+z=1 \\ -y-z=-3 \end{array} $$

Step-by-Step Solution

Answer

The system has no solution; it is inconsistent.

Step 1: Write the System as an Augmented Matrix

Convert the given system of equations into an augmented matrix. The system is: \( x + y = 2 \), \( x + z = 1 \), and \( -y - z = -3 \). The augmented matrix is: \[ \begin{bmatrix} 1 & 1 & 0 & | & 2 \ 1 & 0 & 1 & | & 1 \ 0 & -1 & -1 & | & -3 \end{bmatrix} \].

Step 2: Apply Row Operations to Make Upper Triangular Form

Our goal is to create zeros below the leading coefficients (the pivots) in the first column. Use the row operation \( R_2 = R_2 - R_1 \) to eliminate \( x \) from the second row, giving us \[ \begin{bmatrix} 1 & 1 & 0 & | & 2 \ 0 & -1 & 1 & | & -1 \ 0 & -1 & -1 & | & -3 \end{bmatrix} \].

Step 3: Eliminate the Next Element in the Second Column

We continue by eliminating the element in the third row in the second column. Use the row operation \( R_3 = R_3 + R_2 \) which gives us \[ \begin{bmatrix} 1 & 1 & 0 & | & 2 \ 0 & -1 & 1 & | & -1 \ 0 & 0 & 0 & | & -4 \end{bmatrix} \].

Step 4: Check for Consistency

Notice that the last row now reads \( 0x + 0y + 0z = -4 \), which is an inconsistent row since no combination of variables can equal \(-4\).

Step 5: Conclude the System has No Solution

The presence of an inconsistent row signifies that the system has no solution. Therefore, this system of equations is inconsistent and cannot be solved further.

Key Concepts

Augmented Matrix

To solve a system of equations using Gaussian elimination, one of the first steps involves creating an augmented matrix. An augmented matrix combines the coefficients of the variables from the system of equations with the constants from the right-hand side of each equation. By organizing these into a matrix format, it becomes easier to apply systematic row operations.For instance, given the system:

• \( x + y = 2 \)
• \( x + z = 1 \)
• \( -y - z = -3 \)

This is converted into the augmented matrix form:\[\begin{bmatrix}1 & 1 & 0 & | & 2 \1 & 0 & 1 & | & 1 \0 & -1 & -1 & | & -3\end{bmatrix}\]This matrix showcases the coefficients of \(x, y, \text{and } z\) alongside the constants, separated by a line.

Row Operations

Row operations are the actions we take to simplify the augmented matrix and eventually solve the equations. There are three main types of row operations that can be performed:

• Swapping two rows
• Multiplying all entries of a row by a non-zero constant
• Adding or subtracting the scaled multiple of one row to another row

In this exercise, we used row operations to reach an upper triangular form. For example, to eliminate the \(x\) from the second row, we used the operation \(R_2 = R_2 - R_1\). The resulting matrix was:\[\begin{bmatrix}1 & 1 & 0 & | & 2 \0 & -1 & 1 & | & -1 \0 & -1 & -1 & | & -3\end{bmatrix}\]Further row operations are applied until the matrix reveals the nature of the system.

Inconsistent System

An inconsistent system is one where no solution exists. In the context of our augmented matrix, this is apparent when we derive a row that states a false equation, such as \(0 = -4\).This occurs because, after all feasible row operations, we reach a critical point where the combination of coefficients does not satisfy the corresponding constant term. In our case, the last row of the transformed matrix became:\[0x + 0y + 0z = -4\]Since no values for \(x, y, \text{and } z\) can make this statement true, the system is declared inconsistent and unsolvable.

Upper Triangular Form

A matrix is in upper triangular form when all the entries below the main diagonal are zeros. This form is essential in Gaussian elimination as it facilitates back substitution, a straightforward method to find solutions. Through a series of row operations, we aimed to achieve an upper triangular form by making zeros below the leading coefficients. In our task, however, reaching upper triangular form revealed the inconsistency of the system, evidenced by the row of zeros producing a non-zero constant on the right side of the augmented matrix. Despite achieving part of our goal, the impossibility of a solution became apparent due to the inconsistency rather than a clean triangular shape. This scenario underscores the importance of verifying each transformation step to ensure the process aligns with solving the original system.