Problem 24 - Precalculus: Mathematical for Calculus

Question

Find the complete solution of the linear system, or show that it is inconsistent. $$\left\\{\begin{aligned} -x+2 y+5 z &=4 \\ x-\quad\space\quad2 z &=0 \\ 4 x-2 y-11 z &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent as there is a contradiction in the equations.

Step 1: Write the System in Matrix Form

First, express the system of linear equations in its augmented matrix form. This means writing out the coefficient matrix together with the constants from the right-hand side:\[\begin{bmatrix}-1 & 2 & 5 & | & 4 \1 & 0 & -2 & | & 0 \4 & -2 & -11 & | & 2\end{bmatrix}\]

Step 2: Apply Gaussian Elimination

Use row operations to transform the matrix into row-echelon form. Start with the first row and eliminate the first element under the pivots:1. Add Row 1 to Row 2: \[ R2 = R2 + R1 ightarrow \begin{bmatrix} -1 & 2 & 5 & | & 4 \ 0 & 2 & 3 & | & 4 \ 4 & -2 & -11 & | & 2 \end{bmatrix} \]2. Add 4 times Row 1 to Row 3: \[ R3 = R3 + 4R1 ightarrow \begin{bmatrix} -1 & 2 & 5 & | & 4 \ 0 & 2 & 3 & | & 4 \ 0 & 6 & 9 & | & 18 \end{bmatrix} \]

Step 3: Simplify and Solve for Variables

Continue to simplify the matrix.1. Divide the second row by 2: \[ R2 = \frac{1}{2}R2 ightarrow \begin{bmatrix} -1 & 2 & 5 & | & 4 \ 0 & 1 & 1.5 & | & 2 \ 0 & 6 & 9 & | & 18 \end{bmatrix} \]2. Subtract 6 times the second row from the third row to zero out the second entry of Row 3: \[ R3 = R3 - 6R2 ightarrow \begin{bmatrix} -1 & 2 & 5 & | & 4 \ 0 & 1 & 1.5 & | & 2 \ 0 & 0 & 0 & | & 6 \end{bmatrix} \]

Step 4: Identify Inconsistencies

Observe the rank and consistency of the system.The third row \(0x + 0y + 0z = 6\) represents an equation \(0 = 6\), which is a contradiction. This indicates there is no solution, and the system is inconsistent.

Key Concepts

Linear Systems

A linear system of equations is a collection of two or more linear equations involving the same set of variables. In our example, the system consists of three equations involving the variables \( x \), \( y \), and \( z \). The goal when working with linear systems is to find values for these variables that simultaneously satisfy all the equations in the system. Linear systems can often be solved using algebraic methods, such as substitution or elimination, or by matrix-based techniques like Gaussian elimination. The outcome of solving a linear system is either a set of solutions, indicating one or more combinations of values for the variables that satisfy all equations, or an indication that no solutions exist, making the system inconsistent.

Augmented Matrix

When dealing with linear systems, it’s advantageous to express the system as an augmented matrix. An augmented matrix is a convenient way to organize the coefficients of the system and the constants from each equation in a tabular form. Consider the linear system in our example: - The first row of the matrix corresponds to the equation \(-x + 2y + 5z = 4\).- The second row corresponds to \(x - 2z = 0\).- And the third row corresponds to \(4x - 2y - 11z = 2\).To convert the system into an augmented matrix, list the coefficients from each term in the equations beside each other, and place the constant from the right side of the equation in a final column separated by a line (signifying 'augmented'). This matrix form is especially useful for applying systematic row operations in methods such as Gaussian elimination. By condensing the system into matrix form, it becomes easier to manipulate and solve using algorithmic strategies.

Inconsistent System

An inconsistent system of equations is one that has no solution. This occurs when the equations contradict each other, despite representing linear relationships. In our example, during the Gaussian elimination process, we arrived at a row in the reduced matrix which read \(0x + 0y + 0z = 6\). This equation implies that \(0 = 6\), which is clearly a contradiction as zero can never equal six. Such a contradiction signifies that there is no set of values for the variables that can satisfy all the equations simultaneously. Therefore, the original system of equations is inconsistent. In practice, identifying inconsistencies can save time since it indicates that attempting to determine specific values for variables isn't possible. Thus, recognizing and understanding this concept is vital, as it directs attention towards re-evaluating the initial conditions, or constraints, to potentially adjust them or address errors in the modelling of the problem.