Problem 26 - Algebra and Trigonometry

Question

Find the complete solution of the linear system, or show that it is inconsistent. $$ \left\\{\begin{aligned} x-2 y-3 z &=5 \\ 2 x+y-z &=5 \\ 4 x-3 y-7 z &=5 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent because it leads to a contradiction, implying no solutions exist.

Step 1: Write the system in matrix form

First, express the system of linear equations as an augmented matrix. The system is: \[ \begin{align*} x - 2y - 3z &= 5, \ 2x + y - z &= 5, \ 4x - 3y - 7z &= 5. \end{align*} \] This leads to the augmented matrix: \[ \begin{bmatrix} 1 & -2 & -3 & | & 5 \ 2 & 1 & -1 & | & 5 \ 4 & -3 & -7 & | & 5 \end{bmatrix}. \]

Step 2: Apply row operations to simplify

Our goal is to achieve a row-echelon form. Start by making the coefficients below the first pivot (1 in the first row, first column) into zeros. Perform the row operation \( R_2 = R_2 - 2R_1 \) and \( R_3 = R_3 - 4R_1 \). The matrix becomes: \[ \begin{bmatrix} 1 & -2 & -3 & | & 5 \ 0 & 5 & 5 & | & -5 \ 0 & 5 & 5 & | & -15 \end{bmatrix}. \]

Step 3: Further simplify the matrix

Next, simplify the second and third rows to make the entry in the second row, second column a leading 1. Perform \( R_3 = R_3 - R_2 \) to eliminate one more variable: \[ \begin{bmatrix} 1 & -2 & -3 & | & 5 \ 0 & 5 & 5 & | & -5 \ 0 & 0 & 0 & | & -10 \end{bmatrix}. \]

Step 4: Analyze the resulting matrix

Look at the third row of the matrix \([0, 0, 0 | -10]\), which represents the equation \(0x + 0y + 0z = -10\). This is a contradiction since it implies \(0 = -10\).

Step 5: Conclusion about the system's consistency

Because the system leads to a contradictory statement, the system has no solutions. This means the system is inconsistent.

Key Concepts

Augmented Matrix

In linear algebra, a powerful method to solve systems of linear equations involves using matrices, particularly the augmented matrix. This is essentially an extension of the coefficient matrix, created by appending a column to represent the constants from the equations.
For example, given three equations:

• \(x - 2y - 3z = 5\)
• \(2x + y - z = 5\)
• \(4x - 3y - 7z = 5\)

We form the augmented matrix:\[\begin{bmatrix} 1 & -2 & -3 & | & 5 \ 2 & 1 & -1 & | & 5 \ 4 & -3 & -7 & | & 5 \end{bmatrix}\]Here, the vertical line symbolizes the division between the coefficients of the variables and the constants. This visual representation simplifies the process of applying row operations later.
The augmented matrix approach is beneficial because it streamlines manipulation, making complex systems easier to handle. You'll often use this format for consistency when performing row operations.

Row Operations

Row operations are fundamental techniques in matrix arithmetic, crucial for transforming matrices into a simplified form, particularly the row-echelon form or reduced row-echelon form. They include three basic moves:

• Swapping two rows
• Multiplying a row by a non-zero constant
• Adding or subtracting multiples of one row to another

In our context, applying these operations helps isolate variables one by one. Starting with the given augmented matrix:\[\begin{bmatrix} 1 & -2 & -3 & | & 5 \ 2 & 1 & -1 & | & 5 \ 4 & -3 & -7 & | & 5 \end{bmatrix}\]We perform the row operation \( R_2 = R_2 - 2R_1 \) and \( R_3 = R_3 - 4R_1 \) to simplify the matrix. This yields:\[\begin{bmatrix} 1 & -2 & -3 & | & 5 \ 0 & 5 & 5 & | & -5 \ 0 & 5 & 5 & | & -15 \end{bmatrix}\]Further operations such as \( R_3 = R_3 - R_2 \) continue to simplify the matrix.
These operations aim to clear columns under pivot positions (leading ones), aiming first for zeroes and subsequently for an identity or triangular form.

Inconsistent System

An inconsistent system is characterized by having no solutions due to contradictory equations. Using the row-reduced matrices, we can identify inconsistency by spotting a row that translates into an impossible equation.
In the step-by-step solution, after performing row operations, the matrix becomes:\[\begin{bmatrix} 1 & -2 & -3 & | & 5 \ 0 & 5 & 5 & | & -5 \ 0 & 0 & 0 & | & -10 \end{bmatrix}\]The third row represents the equation:\(0x + 0y + 0z = -10\).
This equation indicates a logical contradiction, as it suggests \(0 = -10\), which is clearly false. A situation like this confirms the system's inconsistency, as there are no values for \(x, y,\) and \(z\) that can satisfy all equations simultaneously. Recognizing such contradictions assists in understanding that no solution exists, allowing students to confidently conclude inconsistency in a system.