Problem 33 - Algebra and Trigonometry

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$ \left\\{\begin{aligned} x-y+3 z &=3 \\ 4 x-8 y+32 z &=24 \\ 2 x-3 y+11 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system of equations is inconsistent.

Step 1: Set Up the Augmented Matrix

Write the system of linear equations in augmented matrix form. The system is:\[ \begin{bmatrix} 1 & -1 & 3 & | & 3 \ 4 & -8 & 32 & | & 24 \ 2 & -3 & 11 & | & 4 \end{bmatrix} \]

Step 2: Perform Row Operations

Use Gaussian elimination to simplify the matrix. Start with the first row and perform operations to create zeros below the pivot element in the first column.Subtract 4 times the first row from the second row:\[R_2 = R_2 - 4R_1 \Rightarrow \begin{bmatrix} 1 & -1 & 3 & | & 3 \ 0 & -4 & 20 & | & 12 \ 2 & -3 & 11 & | & 4 \end{bmatrix} \]Subtract 2 times the first row from the third row:\[R_3 = R_3 - 2R_1 \Rightarrow \begin{bmatrix} 1 & -1 & 3 & | & 3 \ 0 & -4 & 20 & | & 12 \ 0 & -1 & 5 & | & -2 \end{bmatrix} \]

Step 3: Continue Row Operations

Next, eliminate the entry below the pivot in the second column by replacing the third row:Multiply the second row by -1/4:\[R_2 = -1/4 R_2 \Rightarrow \begin{bmatrix} 1 & -1 & 3 & | & 3 \ 0 & 1 & -5 & | & -3 \ \ 0 & -1 & 5 & | & -2 \end{bmatrix} \]Add the second row to the third row:\[R_3 = R_3 + R_2 \Rightarrow \begin{bmatrix} 1 & -1 & 3 & | & 3 \ 0 & 1 & -5 & | & -3 \ 0 & 0 & 0 & | & -5 \end{bmatrix} \]

Step 4: Identify Inconsistency

The final row of the matrix is \([0, 0, 0 | -5]\), which represents the equation \(0x + 0y + 0z = -5\). This implies a contradiction since 0 cannot equal -5, making the system inconsistent.

Key Concepts

Gaussian Elimination

Gaussian Elimination is a powerful technique used to solve systems of linear equations. It involves transforming the system's equations into a row-echelon form, making it easier to extract solutions for the variables. This is achieved by performing a series of row operations to modify the coefficients of the equations. Each step aims to simplify one part of the matrix, working from top to bottom, and left to right.

In essence, it works like this:

• Select a pivot element in the first row and manipulate the rows below to create zeros in the same column under the pivot.
• Move to the next row and column, and repeat the process to create a triangular form.
• Once in row-echelon form, the system can be solved using back substitution.

This method is systematic and reduces the need for rote memorization of solutions, focusing instead on logical steps that consistently bring the equations closer to solvability.

Augmented Matrix

An augmented matrix is an essential concept when working with systems of linear equations. It combines the coefficients of the variables and the constants into a compact and organized format. This form provides a streamlined way of applying operations without losing track of equations.

To create an augmented matrix:

• Write the coefficients of each variable in the columns for each equation.
• Add an additional column for the constants to the right, separated by a line to symbolize equality.

Representing the system as an augmented matrix allows for easier visualization when performing row operations during Gaussian Elimination. It keeps the system of equations neat and helps avoid errors when adjusting equations.

Inconsistent System

An inconsistent system of equations is one that has no solution. This occurs when the equations represent parallel lines that never intersect, or when a contradiction arises, such as having a condition like 0 equating to a non-zero constant as seen in the final row of a matrix.

In the exercise given, the representation \(\begin{bmatrix} 0 & 0 & 0 & | & -5 \end{bmatrix}\) indicates such a contradiction, signaling inconsistency. This row implies that no combination of variable values can satisfy the equation, thus confirming no shared solution exists for the system. Recognizing inconsistency is crucial to avoid wasting time searching for solutions that aren't there.

Row Operations

Row operations are the tools that make Gaussian Elimination possible. Performing row operations consists of three main types:

• Swapping two rows;
• Multiplying a row by a non-zero scalar;
• Adding or subtracting a multiple of one row to another row.

These operations help in simplifying the augmented matrix towards the row-echelon form by systematically zeroing out coefficients below the pivot points.

For example, if you want to create a zero below a pivot, you might multiply the pivot row by a suitable number and subtract it from the row below. These steps transform the system without changing its solution set, which allows for accurate determination of whether the system is consistent and what those solutions might be if they exist.