Published by: Dikshya
Published date: 18 Jul 2023
The existence of solutions for a linear set of equations depends on the properties of the system. In general, a system of linear equations has three possibilities:
1. Consistent System:
- The system has at least one solution.
- The equations in the system can be satisfied simultaneously.
- The system may have a unique solution or infinitely many solutions.
- The number of equations in the system is equal to the number of variables.
- The equations are independent, meaning that no equation can be obtained by adding or subtracting other equations.
- The solution(s) represent the point(s) of intersection of the lines, planes, or hyperplanes defined by the equations.
- The system is solvable using various methods such as substitution, elimination, matrix representation, or Gaussian elimination.
- Each equation provides additional information about the variables and their relationships.
- The solution(s) can be verified by substituting them back into the original equations, which should satisfy all equations.
- In a consistent system, the number of independent equations is equal to the number of variables, ensuring a well-determined solution.
2. Inconsistent System:
- The system of equations cannot be solved simultaneously.
- The equations do not have a common solution.
- The lines, planes, or hyperplanes defined by the equations do not intersect.
- The system has no solution.
- The equations are contradictory.
- The equations represent parallel lines, planes, or hyperplanes that never meet.
- Any attempt to find a solution to the system leads to a contradiction.
- The coefficient matrix of the system is singular, meaning its determinant is zero.
- Geometrically, the equations represent objects that do not overlap.
- The system violates one or more equations, making it impossible to find a solution that satisfies all of them.
3. Degenerate System:
- A degenerate system has infinitely many solutions.
- The equations in a degenerate system are dependent.
- Dependent equations in a degenerate system represent the same line, plane, or hyperplane.
- Any point on the common line, plane, or hyperplane is a solution to the system.
- A degenerate system does not have a unique solution.
- The number of independent equations in a degenerate system is less than the number of variables.
- The excess variables in a degenerate system can be chosen freely to represent the infinite solutions.
- A degenerate system can be represented by fewer equations than variables, as long as the equations are dependent.
- The rank of the coefficient matrix in a degenerate system is less than the number of variables.
- Determining the specific solutions of a degenerate system requires expressing the variables in terms of the free variables.