Analyzing The Consistency Of A System Of Linear Equations
In linear algebra, determining the consistency of a system of equations is a fundamental task. A system of equations is considered consistent if it has at least one solution, meaning there exists a set of values for the variables that satisfy all equations simultaneously. Conversely, a system is inconsistent if it has no solution. This article delves into the methods for assessing the consistency of a system of linear equations, focusing on the specific example:
2x + 3y + 4z = 1 x + 5y + 7z = 15 8x + 13y + 12z = 25
To analyze the consistency, we'll employ techniques such as Gaussian elimination and matrix representation to determine if the system possesses a unique solution, infinitely many solutions, or no solution.
Methods for Determining Consistency
Several methods can be used to analyze the consistency of a system of linear equations. Two primary approaches are:
- Gaussian Elimination (Row Reduction): This method involves transforming the system's augmented matrix into row-echelon form or reduced row-echelon form. By observing the resulting matrix, we can determine the system's nature.
- Matrix Representation and Rank: Expressing the system in matrix form (AX = B) allows us to analyze the rank of the coefficient matrix (A) and the augmented matrix ([A|B]). Comparing these ranks reveals the system's consistency.
Gaussian Elimination (Row Reduction)
Gaussian elimination, also known as row reduction, is a powerful technique for solving systems of linear equations and determining their consistency. This method involves performing elementary row operations on the augmented matrix of the system to transform it into either row-echelon form or reduced row-echelon form. The elementary row operations are:
- Swapping two rows.
- Multiplying a row by a non-zero scalar.
- Adding a multiple of one row to another row.
The augmented matrix for our system is:
[ 2 3 4 | 1 ]
[ 1 5 7 | 15 ]
[ 8 13 12 | 25 ]
Let's apply Gaussian elimination:
- Swap row 1 and row 2:
[ 1 5 7 | 15 ]
[ 2 3 4 | 1 ]
[ 8 13 12 | 25 ]
- Replace row 2 with row 2 - 2 * row 1, and replace row 3 with row 3 - 8 * row 1:
[ 1 5 7 | 15 ]
[ 0 -7 -10 | -29 ]
[ 0 -27 -44 | -95 ]
- Multiply row 2 by -1/7:
[ 1 5 7 | 15 ]
[ 0 1 10/7 | 29/7 ]
[ 0 -27 -44 | -95 ]
- Replace row 3 with row 3 + 27 * row 2:
[ 1 5 7 | 15 ]
[ 0 1 10/7 | 29/7 ]
[ 0 0 -38/7 | -38/7 ]
- Multiply row 3 by -7/38:
[ 1 5 7 | 15 ]
[ 0 1 10/7 | 29/7 ]
[ 0 0 1 | 1 ]
The matrix is now in row-echelon form. We can further reduce it to reduced row-echelon form, but for consistency analysis, this form is sufficient. We can observe that there are no rows of the form [0 0 0 | c] where c is non-zero. This indicates that the system is consistent and has a unique solution.
Matrix Representation and Rank
Another approach to determine the consistency of a system of linear equations involves representing the system in matrix form and analyzing the ranks of the coefficient matrix and the augmented matrix. The system can be written as AX = B, where:
- A is the coefficient matrix.
- X is the column vector of variables.
- B is the column vector of constants.
For our system:
A = [ 2 3 4 ]
[ 1 5 7 ]
[ 8 13 12 ]
X = [ x ]
[ y ]
[ z ]
B = [ 1 ]
[ 15 ]
[ 25 ]
The augmented matrix [A|B] is:
[ 2 3 4 | 1 ]
[ 1 5 7 | 15 ]
[ 8 13 12 | 25 ]
The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix. We can determine the rank by performing row operations to bring the matrix into row-echelon form and counting the number of non-zero rows.
From the Gaussian elimination process in the previous section, we already have the row-echelon form of the augmented matrix:
[ 1 5 7 | 15 ]
[ 0 1 10/7 | 29/7 ]
[ 0 0 1 | 1 ]
The rank of the augmented matrix [A|B] is 3, as there are three non-zero rows. Now, we need to find the rank of the coefficient matrix A. We can take the first three columns of the row-echelon form and consider it as a matrix:
[ 1 5 7 ]
[ 0 1 10/7 ]
[ 0 0 1 ]
The rank of the coefficient matrix A is also 3, as there are three non-zero rows. According to the Rank Theorem, a system of linear equations AX = B is consistent if and only if the rank of A is equal to the rank of the augmented matrix [A|B]. Furthermore:
- If rank(A) = rank([A|B]) = number of variables, the system has a unique solution.
- If rank(A) = rank([A|B]) < number of variables, the system has infinitely many solutions.
- If rank(A) < rank([A|B]), the system is inconsistent and has no solution.
In our case, rank(A) = rank([A|B]) = 3, which is equal to the number of variables (x, y, z). Therefore, the system has a unique solution.
Determining the Solution
Since we've established that the system has a unique solution, we can find it using back-substitution from the row-echelon form of the augmented matrix:
[ 1 5 7 | 15 ]
[ 0 1 10/7 | 29/7 ]
[ 0 0 1 | 1 ]
From the last row, we have z = 1. Substituting z into the second row:
y + (10/7) * 1 = 29/7 y = 29/7 - 10/7 y = 19/7
Substituting y and z into the first row:
x + 5 * (19/7) + 7 * 1 = 15 x + 95/7 + 7 = 15 x = 15 - 7 - 95/7 x = 8 - 95/7 x = (56 - 95) / 7 x = -39/7
Thus, the unique solution is x = -39/7, y = 19/7, and z = 1.
Conclusion: Consistency Analysis and Solution
In conclusion, by applying Gaussian elimination and analyzing the ranks of the coefficient and augmented matrices, we have determined that the given system of linear equations:
2x + 3y + 4z = 1 x + 5y + 7z = 15 8x + 13y + 12z = 25
is consistent and has a unique solution. The solution is x = -39/7, y = 19/7, and z = 1. The use of row reduction and rank analysis provides a systematic approach to determine the consistency and nature of solutions for any system of linear equations.
This comprehensive analysis underscores the importance of matrix methods in solving and understanding systems of linear equations, making it a core concept in linear algebra and applicable in various fields such as engineering, physics, and computer science.
