Problem 47 - Intermediate Algebra

Question

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. $$ \left\\{\begin{array}{l} 2 x+y-2 z=6 \\ 4 x-y+z=-1 \\ 6 x-2 y+3 z=-5 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations is inconsistent and has no solution.

Step 1: Write the System in Matrix Form

The given system of linear equations can be represented as a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constant matrix. For the given system:\[A = \begin{bmatrix} 2 & 1 & -2 \ 4 & -1 & 1 \ 6 & -2 & 3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 6 \ -1 \ -5 \end{bmatrix}\]

Step 2: Determine if the Solution Exists Using Augmented Matrix

To find the solution, we examine the augmented matrix \( [A|\mathbf{b}] \) and use Gaussian elimination to reduce it to row-echelon form:\[[A|\mathbf{b}] = \begin{bmatrix} 2 & 1 & -2 & | & 6 \ 4 & -1 & 1 & | & -1 \ 6 & -2 & 3 & | & -5 \end{bmatrix}\]We will perform row operations to simplify this matrix.

Step 3: Perform Row Operations

First, swap Row 1 and Row 2 to get:\[\begin{bmatrix} 4 & -1 & 1 & | & -1 \ 2 & 1 & -2 & | & 6 \ 6 & -2 & 3 & | & -5 \end{bmatrix}\]Next, we eliminate the first column below the pivot:- Replace Row 2 with (Row 2) - 0.5 * (Row 1)- Replace Row 3 with (Row 3) - 1.5 * (Row 1)After performing these operations, the matrix becomes:\[\begin{bmatrix} 4 & -1 & 1 & | & -1 \ 0 & 1.5 & -2.5 & | & 6.5 \ 0 & -0.5 & 1.5 & | & -3.5 \end{bmatrix}\]

Step 4: Continue Gaussian Elimination

To further simplify, eliminate entries below the pivot in Row 2:- Replace Row 3 with (Row 3) + (1/3) * (Row 2)Now the matrix is:\[\begin{bmatrix} 4 & -1 & 1 & | & -1 \ 0 & 1.5 & -2.5 & | & 6.5 \ 0 & 0 & 0 & | & -(7.2) \end{bmatrix}\]

Step 5: Analyze the Row-Echelon Form

The last row of the matrix is \([ 0 \, 0 \, 0 \, | \, -7.2 ]\), indicating that this system is inconsistent since it implies the false statement \(0 = -7.2\). Hence, the system of equations has no solutions, as there is a contradiction in the equations.

Key Concepts

Gaussian elimination

Gaussian elimination is a powerful mathematical method used to solve systems of linear equations. It is designed to systematically eliminate variables from equations by using row operations. The goal is to transform a matrix representing the system into row-echelon form, where the matrix becomes easier to solve.
You can perform Gaussian elimination by applying these steps:

• Identify the pivot, which is the first non-zero element in a row from left to right.
• Use row operations to make all elements below the pivot in the same column zero.
• Repeat the process for the subsequent rows, moving from left to right, and top to bottom.

Gaussian elimination simplifies complicated equations, allowing us to detect inconsistencies or solutions precisely.

augmented matrix

An augmented matrix provides a convenient method to organize and solve systems of linear equations. It combines both the coefficients of the variables and the constant terms into one matrix. The augmented matrix can be visually divided into two sections:

• One part contains the coefficient matrix of the variables.
• The other part consists of the constant terms as a separate column on the right.

A clear representation of an augmented matrix for a system can be written as \([A|\mathbf{b}]\), where \[A\] is the coefficient matrix, and \(\mathbf{b}\) represents the constants. Augmented matrices simplify the process of applying row operations and eventually solving for the variables through methods like Gaussian elimination.

inconsistent system

An inconsistent system of equations is when no solutions exist for the given equations. In the context of matrix equations, an inconsistency is typically revealed during Gaussian elimination. Specifically, this occurs when a row in the row-echelon form results in an impossible equation, for example:

• A row of the form \[0 \, 0 \, 0 \, | \, c\], with \c eq 0, leads to a contradiction because it implies the false statement \(0 = c\).

A system being inconsistent means that there is no set of values for the variables that would satisfy all the equations simultaneously. Recognizing an inconsistent system is crucial for understanding that no solution exists and for preventing futile further calculations.

row-echelon form

Row-echelon form is an important step in solving and understanding systems of linear equations. It is a form where a matrix has been transformed to a step-like shape via row operations. The key characteristics of the row-echelon form include:

• All non-zero rows are above any rows of all zeros.
• The leading entry (also known as pivot) of a non-zero row is always to the right of the leading entry of the row above it.
• All elements below a pivot are zero.

Achieving a row-echelon form through Gaussian elimination makes it easier to solve the system, as it reveals potential inconsistencies or allows for back substitution to find the solution of the system. Understanding how to interpret and manipulate row-echelon form is essential in linear algebra.