Problem 21 - Algebra A Combined Function

Question

Solve each system of linear equations using matrices. See Examples 1 through 3. $$ \left\\{\begin{array}{rr} 4 x-y+2 z= & 5 \\ 2 y+z= & 4 \\ 4 x+y+3 z= & 10 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations is inconsistent, so there is no solution.

Step 1: Write the system as a matrix equation

The system of equations can be written in the form of a matrix equation \( A \mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the column matrix for variables, and \( \mathbf{b} \) is the constant matrix. \[A = \begin{bmatrix} 4 & -1 & 2 \ 0 & 2 & 1 \ 4 & 1 & 3 \end{bmatrix}\]\[\mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}\]\[\mathbf{b} = \begin{bmatrix} 5 \ 4 \ 10 \end{bmatrix} \] These matrices represent the system of equations.

Step 2: Augment the matrix

Form the augmented matrix \( [A|\mathbf{b}] \) by combining matrix \( A \) and vector \( \mathbf{b} \), which includes the coefficients of the variables and the constants on the right side of the equations.\[[A|\mathbf{b}] = \begin{bmatrix} 4 & -1 & 2 & | & 5 \ 0 & 2 & 1 & | & 4 \ 4 & 1 & 3 & | & 10 \end{bmatrix}\]

Step 3: Apply row operations to obtain row-echelon form

Use Gaussian elimination to simplify the augmented matrix to row-echelon form. Start with the first row as your pivot row:1. Subtract the first row from the third row to eliminate the \( x \) term in the third equation: \[ R_3 = R_3 - R_1: \begin{bmatrix} 4 & 1 & 3 & | & 10 \end{bmatrix} - \begin{bmatrix} 4 & -1 & 2 & | & 5 \end{bmatrix} = \begin{bmatrix} 0 & 2 & 1 & | & 5 \end{bmatrix} \] Augmented matrix is now: \[ \begin{bmatrix} 4 & -1 & 2 & | & 5 \ 0 & 2 & 1 & | & 4 \ 0 & 2 & 1 & | & 5 \end{bmatrix} \]2. Subtract the second row from the third: \[ R_3 = R_3 - R_2: \begin{bmatrix} 0 & 2 & 1 & | & 5 \end{bmatrix} - \begin{bmatrix} 0 & 2 & 1 & | & 4 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & | & 1 \end{bmatrix} \] The system is now inconsistent because the last row implies \( 0x + 0y + 0z = 1 \), which is not possible.

Key Concepts

Matrix Equation

A matrix equation is a way to represent a system of linear equations in a concise matrix form. It involves three matrices:

• Coefficient Matrix \(A\)
• Variable Matrix \(\mathbf{x}\)
• Constant Matrix \(\mathbf{b}\)

The matrix equation is written as \(A \mathbf{x} = \mathbf{b}\). Here, \(A\) contains the coefficients of all the variables, \(\mathbf{x}\) is the matrix with the variables of the system \((x, y, z, \text{etc.})\), and \(\mathbf{b}\) is the matrix of constants on the other side of the equations. This method streamlines the process of solving systems of equations.
By writing the system as a matrix equation, the problem changes from manipulating multiple separate equations to handling a single, unified expression in matrix form. This makes operations systematic and allows the use of multiple matrix-solving techniques like Gaussian elimination.

Gaussian Elimination

Gaussian elimination is a technique used to simplify a system of linear equations to an easily interpretable form. The main goal is to transform the matrix into a row-echelon form where the solutions become obvious. Here's how the process works:

• Use row operations to create zeros below each pivot element (first non-zero number from the left in a row).
• Perform operations until the matrix reaches a step-like structure.
• Continue these operations until it's straightforward to solve for the variables.

This method is effective because it systematically reduces the complexity of a system, eventually making it simple to back-substitute and find solutions for the variables. It's like tidying up a messy room - you handle one piece of the puzzle at a time until it's all sorted. The result is either an easy system ready for back substitution or, in some cases, a clearer view that reveals whether the system is inconsistent.

Inconsistent System

An inconsistent system occurs when there are no solutions that satisfy all the equations simultaneously. During Gaussian elimination, you might encounter a row that looks like this:

• \(0x + 0y + 0z = c\) where \(c\) is a non-zero value

This situation is logically impossible, as it suggests something untrue, like 0 equating to 1, which can't happen. In the context of our exercise, when simplifying the matrix, the final step results in \(0x + 0y + 0z = 1\). Such contradictions indicate there's no possible combination of \(x\), \(y\), and \(z\) that can solve the equations together.
An inconsistent system usually indicates that the original equations contradict each other in some manner, meaning they cannot all be true at once. Understanding this term helps recognize when a system has no solutions, an essential part of solving such systems.

Augmented Matrix

The augmented matrix is a combination of the coefficient matrix \(A\) and the constant matrix \(\mathbf{b}\). It arranged side by side to form a new matrix \([A|\mathbf{b}]\). The major advantage of using an augmented matrix lies in its simplicity in handling operations:

• Makes it straightforward to use row operations like scaling, row addition, and row swapping.
• Incorporates all information from the system of equations in one neat package.
• Simplifies visualization of the entire system, including relationships between variables and constants.

The use of an augmented matrix allows the freedom to manipulate entire rows as units, streamlining the process of finding solutions or identifying inconsistencies within the system. It elegantly encapsulates the system, helping to align operations with the ultimate goal of solving or exploring the nature of the system.