Problem 10 - Contemporary Precalculus

Question

In Exercises \(9-12\), the reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of the system. $$\left(\begin{array}{ccccc} 1 & 0 & 0 & 0 & -1 \\ 0 & 1 & 0 & 0 & 3 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right)$$

Step-by-Step Solution

Answer

Question: Determine if the given matrix has a solution for the system of equations and provide the solution if possible. Matrix: [1 0 0 -1] [0 1 0 3] [0 0 1 1] [0 0 0 1] Answer: The given matrix represents an inconsistent system of equations that has no solution.

Step 1: Identify the given matrix in the form of linear equations

The given matrix can be written in the form of the following linear system of equations: $$x = -1$$ $$y = 3$$ $$z = 1$$ $$0 = 1$$

Step 2: Check for inconsistencies

As we can see, the fourth equation indicates an inconsistency in the system, as it states that 0 = 1, which is not true.

Step 3: Determine the solution

Due to the inconsistency in the fourth equation, we can conclude that this system has no solution. The equations are inconsistent, and there is no possible solution for this system.

Key Concepts

Reduced Row Echelon Form

Learning about linear algebra often involves understanding different matrix forms. One of the most fundamental forms is the Reduced Row Echelon Form (RREF). This form is crucial for solving systems of linear equations. When a matrix is in RREF, it has a specific structure that makes it easy to interpret:

• Every leading entry (the first non-zero number from the left, in a row) is 1.
• Each leading 1 is the only non-zero number in its column.
• The leading 1 of each row is to the right of the leading 1 in the row above.
• Rows of all zeros, if there are any, are at the bottom of the matrix.

These properties help in quickly identifying the solution of the system of equations represented by the matrix.

Inconsistent System

The concept of an inconsistent system is vital when handling systems of equations. An inconsistent system is one that has no solutions. The reason lies within its structure. When transformed into equations, if you encounter a statement that is false, like 0 = 1, the system is inconsistent.

This occurs because there is a contradiction among the equations. They might represent parallel lines that never intersect, or have logical contradictions. Detecting inconsistencies early saves time since it indicates that solving the system further will not yield a valid result.

Augmented Matrix

An augmented matrix is especially useful in linear algebra. It combines the coefficients of the variables with the constants from the equations they relate to. In short, it is a compact way to express a system of equations. To construct an augmented matrix, follow these steps:

• Extract the coefficients of each variable from each equation.
• Place these coefficients into a matrix where each row corresponds to a single equation.
• Extend the matrix to include the constants from each equation to the far right.

This format makes it possible to apply row operations systematically, leading either to solutions or revealing inconsistencies in the system.

System of Equations

In mathematics, a system of equations is a set of equations with multiple variables. The goal is usually to find values for these variables that satisfy all of the equations simultaneously. Systems of equations can:

• Have a single solution, making them consistent and independent.
• Have infinitely many solutions, making them consistent and dependent.
• Have no solution, making them inconsistent.

Understanding the nature of these systems helps in efficiently determining solutions or identifying impossibilities through methods like Gaussian elimination or interpreting the solutions directly from the reduced row echelon form. These insights allow for solving real-world problems that can be modeled mathematically through systems of equations.