Proving the Consistency of Linear Equations: A Comprehensive Guide
Understanding the consistency of linear equations is crucial for solving and analyzing systems of linear equations. A linear equation is deemed consistent if there exists at least one set of variables that can satisfy the equation. In this context, we primarily deal with systems of linear equations, not individual equations.
What is a Consistent System of Linear Equations?
A system of linear equations is considered consistent if it has at least one solution. In other words, there is a combination of variable values that satisfies all equations simultaneously. Conversely, an inconsistent system has no such combination of values that solve all equations at once.
Key Concept: Homogeneous vs Non-homogeneous Systems
A homogeneous system of linear equations is one where all constants on the right-hand side (RHS) of the equations are zero. Homogeneous systems are always consistent because adding an extra column of zeros to the augmented matrix does not increase its rank. Conversely, non-homogeneous systems can be inconsistent if they lead to a contradiction, such as 0 1.
Proving Consistency: Rank of Matrices
To prove that a given system of linear equations is consistent, one effective method is to examine the ranks of the coefficient matrix and the corresponding augmented matrix.
Step 1: Augmented Matrix
Start by forming the coefficient matrix (A) and the augmented matrix (A|B). The coefficient matrix A contains the coefficients of the variables, while the augmented matrix A|B includes the constants from the RHS of the equations as an additional column.
Step 2: Row Echelon Form
Row reduce the augmented matrix to its row-echelon form using elementary row operations. This process does not change the consistency of the system. If the ranks of the coefficient matrix and the augmented matrix are equal, the system is consistent. If they differ, the system is inconsistent.
Example of a Consistent System
Consider the system of equations:
2x 3y 54x 6y 10
The augmented matrix for this system is:
[2 3 | 5;4 6 | 10]
Row reduce to row-echelon form:
[1 1.5 | 2.5;0 0 | 0]
The rank of both matrices is 1, so the system is consistent.
Example of an Inconsistent System
Consider the system of equations:
2x 3y 54x 6y 15
The augmented matrix for this system is:
[2 3 | 5;4 6 | 15]
Row reduce to row-echelon form:
[1 1.5 | 2.5;0 0 | 0]
The rank of the coefficient matrix is 1, while the rank of the augmented matrix is 2, indicating inconsistency.
Consistency vs Inconsistency: Visual Explanation
Algebraically, consistency is shown when you have one or an infinite number of points (a solution or solutions) that satisfy the system of equations. If the lines intersect at a point or coincide, the system is consistent. If the lines are parallel and never intersect, the system is inconsistent.
Conclusion
Proving the consistency of a linear system involves checking the ranks of the coefficient and augmented matrices. This method is highly effective and simplifies the process of solving and analyzing systems of linear equations. Whether dealing with a homogeneous or non-homogeneous system, understanding the consistency of linear equations is fundamental to solving problems in various mathematical and real-world applications.