Problem 55 - A Graphical Approach to Precalculus with Limits

Question

Use row operations on an augmented matrix to solve each system of equations. Round to nearest thousandth when appropriate. $$\begin{aligned} x+y-2 z &=-6 \\ x-y+z &=4 \\ 2 x-z &=-1 \end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution and is inconsistent.

Step 1: Write the Augmented Matrix

Start by representing the system of equations in augmented matrix form. The system of equations is:\[\begin{align*} x+y-2z &=-6 \x-y+z &=4 \2x-z &=-1\end{align*}\]The corresponding augmented matrix is:\[\begin{bmatrix}1 & 1 & -2 & | & -6 \1 & -1 & 1 & | & 4 \2 & 0 & -1 & | & -1\end{bmatrix}\]

Step 2: Eliminate the First Column Entries Below the First Row

To eliminate the entries in the first column below the first row, perform the row operations. Replace the second row \(R_2\) by \(R_2 - R_1\):\[R_2 = \begin{bmatrix} 1 & -1 & 1 & | & 4 \end{bmatrix} - \begin{bmatrix} 1 & 1 & -2 & | & -6 \end{bmatrix} =\begin{bmatrix} 0 & -2 & 3 & | & 10 \end{bmatrix}\]Now, replace the third row \(R_3\) by \(R_3 - 2R_1\):\[R_3 = \begin{bmatrix} 2 & 0 & -1 & | & -1 \end{bmatrix} - 2\begin{bmatrix} 1 & 1 & -2 & | & -6 \end{bmatrix} =\begin{bmatrix} 0 & -2 & 3 & | & 11 \end{bmatrix}\]The matrix now becomes:\[\begin{bmatrix}1 & 1 & -2 & | & -6 \0 & -2 & 3 & | & 10 \0 & -2 & 3 & | & 11\end{bmatrix}\]

Step 3: Further Eliminate and Simplify

Recognize that the third row is a multiple of the second row. Subtract the second row from the third row to obtain:\[R_3 = \begin{bmatrix} 0 & -2 & 3 & | & 11 \end{bmatrix} - \begin{bmatrix} 0 & -2 & 3 & | & 10 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & | & 1 \end{bmatrix}\]This reveals an inconsistency (as the third row indicates the false statement 0 = 1), suggesting there are no solutions. Thus, the system is inconsistent.

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used in linear algebra to solve systems of equations. Essentially, it translates a system of equations into a matrix form, making it easier to apply various matrix operations.
An augmented matrix includes the coefficients of the variables and the constants from the equations.Here's why it's useful:

• The augmented matrix provides a compact, visual representation of equations that simplifies the process of solving them using row operations.
• It allows us to apply algorithms, like Gaussian elimination, to systematically reduce the system.
• By manipulating rows, we can transform a complex system into a simpler one, or even identify inconsistency, as in the given example.

Let's consider our exercise example, which led to the matrix:\[\begin{bmatrix}1 & 1 & -2 & | & -6 \1 & -1 & 1 & | & 4 \2 & 0 & -1 & | & -1\end{bmatrix}\]In this form, the matrix allows us to use row operations to move towards finding solutions, or as here, determine an inconsistency.

System of Equations

A system of equations is a set of equations with multiple variables that we want to solve simultaneously. These systems arise in various real-world contexts, such as engineering, physics, and economics, where interconnected conditions must be satisfied at the same time.
The goal when working with these systems is to find values for the variables that satisfy all equations in the system.Imagine each equation as a line, and the solution is where these lines intersect. Depending on the number and orientation of these lines:

• There may be one unique solution where all lines intersect at one point.
• There could be infinitely many solutions if the lines are the same (infinite intersections).
• There might be no solution if the lines don't intersect at all.

In our exercise's case, represented as the system:\[x + y - 2z = -6 \x - y + z = 4 \2x - z = -1\]When translated to a matrix and row operations are applied, it revealed an inconsistency, indicating no possible solution exists where all three equations intersect.

Inconsistent System

An inconsistent system occurs when there is no solution that satisfies all equations in a system. This is often revealed during matrix operations, like when transforming the system of equations into row-echelon form.
In such instances, the row operations highlight conflicting equations or impossible scenarios.For example, in the provided solution, the matrix was ultimately simplified to show:\[\begin{bmatrix}0 & 0 & 0 & | & 1\end{bmatrix}\]This row indicates that we would need something impossible (0 = 1) to solve the system, showcasing an inherent contradiction.Here are a few key points to remember about inconsistent systems:

• They often occur when there is a logical contradiction within the equations.
• The augmented matrix approach exposes these contradictions in a systematic way.
• Understanding the nature of inconsistency helps us look back and verify if the equations were initially set up correctly or if they represent an impossible scenario.

In summary, identifying an inconsistency through the use of matrices and row operations saves time and clearly proves that the system of equations cannot have solutions that meet all given conditions.