Question

The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use \(x, y ;\) or \(x, y, z ;\) or \(x_{1}, x_{2}, x_{3}, x_{4}\) as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. $$ \left[\begin{array}{llll|l} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 & 2 \\ 0 & 0 & 1 & 3 & 0 \end{array}\right] $$

Short answer

The system is consistent with the solution: \(x_{1} = 1\), \(x_{2} = 2 - 2x_{4}\), \(x_{3} = -3x_{4}\).

Step-by-step solution

Identify Variables

Assign variables to each column of the coefficient matrix. Let the variables be: Column 1: \(x_{1}\) Column 2: \(x_{2}\) Column 3: \(x_{3}\) Column 4: \(x_{4}\)

Write the System of Equations

Translate each row of the augmented matrix into an equation: Row 1: \(1x_{1} + 0x_{2} + 0x_{3} + 0x_{4} = 1\) simplifies to \(x_{1} = 1\) Row 2: \(0x_{1} + 1x_{2} + 0x_{3} + 2x_{4} = 2\) simplifies to \(x_{2} + 2x_{4} = 2\) Row 3: \(0x_{1} + 0x_{2} + 1x_{3} + 3x_{4} = 0\) simplifies to \(x_{3} + 3x_{4} = 0\)

Analyze Consistency

There are no rows of the form \(0\ 0\ 0\ 0\ | b\) where \(b\) is non-zero, indicating that the system is consistent.

Solve the Equations

First, solve the equations involving the leading variables: From Row 3: \(x_{3} = -3x_{4}\) From Row 2: \(x_{2} = 2 - 2x_{4}\) From Row 1: \(x_{1} = 1\) Therefore, the general solution for the variables is \(x_{1} = 1\), \(x_{2} = 2 - 2x_{4}\) and \(x_{3} = -3x_{4}\).

Key concepts

System of Linear Equations

A system of linear equations involves multiple linear equations grouped together.
Each equation in the system represents a line, plane, or hyperplane depending on the number of variables.
For instance, in a system involving variables like \(x_1, x_2, x_3,\) and \(x_4\), we have multiple equations that these variables need to satisfy simultaneously.
The goal is to find values for the variables such that all equations hold true at the same time.

Inconsistent Systems

An inconsistent system of linear equations has no solution at all.
This happens when the equations contradict each other.
In matrix form, this is usually indicated by having a row in the reduced row echelon form (RREF) that looks like \([0ewline 0ewline 0ewline 0 | b]\) where \(b\) is not zero (\(b e 0\)).
Such a row translates to an equation where 0 equals a non-zero number, a clear contradiction.

Consistent Systems

A consistent system of linear equations is one that has at least one solution.
Depending on the system, this solution can be unique or there can be infinitely many solutions.
In the given exercise, the matrix provided is in reduced row echelon form (RREF), and without any rows indicating a contradiction (like \([0ewline 0ewline 0ewline 0 | b]\)), the system is consistent.

General Solution

The general solution of a system of linear equations expresses the solutions in terms of free variables, if any.
In the reduced row echelon form (RREF), the leading variables are those that correspond to the first non-zero entry in each row.
Free variables do not correspond to leading entries and can take on arbitrary values.

From the solution given, we know:
• \(x_{1} = 1\)
• \(x_{2} = 2 - 2x_{4}\)
• \(x_{3} = -3x_{4}\)

Here, \(x_4\) is a free variable and can take any value, providing us with infinitely many solutions based on the value of \(x_4\). The rest of the variables depend on \(x_4\).