Problem 26 - Precalculus

Question

In Exercises \(9-26\), put each system of linear equations into triangular form and solve the system if poesible. Classify each system as consistent independent, consistent dependent, or inconsistent. $$ \left\\{\begin{aligned} x_{1}-x_{2}-5 x_{3}+3 x_{4} &=-1 \\ x_{1}+x_{2}+5 x_{3}-3 x_{4} &=0 \\ x_{2}+5 x_{3}-3 x_{4} &=1 \\ x_{1}-2 x_{2}-10 x_{3}+6 x_{4} &=-1 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent with no solutions.

Step 1: Write the Augmented Matrix

Begin by writing the augmented matrix for the given system of equations. The system:\[\begin{align*}x_{1} - x_{2} - 5 x_{3} + 3 x_{4} &= -1 \x_{1} + x_{2} + 5 x_{3} - 3 x_{4} &= 0 \x_{2} + 5 x_{3} - 3 x_{4} &= 1 \x_{1} - 2 x_{2} - 10 x_{3} + 6 x_{4} &= -1\end{align*}\]translates into the matrix:\[\begin{bmatrix}1 & -1 & -5 & 3 & | & -1 \1 & 1 & 5 & -3 & | & 0 \0 & 1 & 5 & -3 & | & 1 \1 & -2 & -10 & 6 & | & -1\end{bmatrix}\]

Step 2: Perform Row Operations for Triangular Form

Subtract Row 1 from Row 2, Row 1 from Row 4 to eliminate variables:\[\begin{bmatrix}1 & -1 & -5 & 3 & | & -1 \0 & 2 & 10 & -6 & | & 1 \0 & 1 & 5 & -3 & | & 1 \0 & -1 & -5 & 3 & | & 0\end{bmatrix}\]Add Row 3 to Row 4:\[\begin{bmatrix}1 & -1 & -5 & 3 & | & -1 \0 & 2 & 10 & -6 & | & 1 \0 & 1 & 5 & -3 & | & 1 \0 & 0 & 0 & 0 & | & 1\end{bmatrix}\]

Step 3: Analyze the Matrix to Determine Consistency

The last row represents the equation \(0 = 1\), which is a contradiction, indicating that the system has no solutions.

Step 4: Classify the System

Since there is a contradiction in the matrix, the system is classified as inconsistent.

Key Concepts

Triangular Form

In solving systems of linear equations, one helpful technique is to convert the system into triangular form. This involves arranging the coefficients of the variables so that in each row, after the first, has one more leading zero than the previous row. This results in a matrix resembling an upper triangular matrix when viewed in its augmented form.

Converting to a triangular form helps simplify the solution process through approaches like back substitution. By focusing on one variable at a time from the bottom row upwards, each equation becomes simpler as the triangular form eliminates variables strategically.

Augmented Matrix

In linear algebra, an augmented matrix is a compact representation of a system of linear equations. It includes the coefficients of the variables and the constants from the right-hand side of the equations.

For example, consider the system of linear equations given in the problem. We can express these equations in a matrix format:

• The first column represents coefficients of \(x_1\).
• The second column represents coefficients of \(x_2\).
• The third column represents coefficients of \(x_3\).
• The fourth column represents coefficients of \(x_4\).
• The constants from the equations are placed in the last column, separated by a "|".

This structured matrix consolidates the information of the equations for easier manipulation, especially when performing row operations.

Consistent and Inconsistent Systems

In the context of linear equations, a system can be classified as either consistent or inconsistent.

• A consistent system has at least one solution. It either has a unique solution, known as independent, or infinitely many solutions, known as dependent.
• An inconsistent system does not have any solutions. In practical terms, this occurs when you reach a situation where a false statement, such as \(0 = 1\), is derived from the system during calculations.

In our problem, the operation led to a row resembling \([0, 0, 0, 0 | 1]\), translating to the equation \(0 = 1\). This indicates no solution exists, making the system inconsistent.

Row Operations

Row operations are essential tools used to manipulate matrices when solving linear systems. They include:

• Swapping two rows.
• Multiplying a row by a non-zero constant.
• Adding or subtracting the multiple of one row to another, called row addition.

In our exercise, row operations were employed to simplify the augmented matrix into triangular form, making it easier to analyze and solve.

The strategic application of these operations helps in isolating individual variables or, like in our case, indicating inconsistencies in the system which point out that no solution is possible.