How to Solve Systems of Linear Equations
A system of linear equations can be written compactly as the matrix equation Ax = B, where A is the coefficient matrix, x is the vector of unknown variables, and B is the vector of constants. Performing Gauss-Jordan elimination on the augmented matrix [A | B] is the standard algebraic method to solve for the vector x.
Classifying the Solution Set
When you reduce the augmented matrix of a system of equations, you will arrive at one of three distinct algebraic outcomes:
1. Consistent System with a Unique Solution
This occurs when every column of the coefficient matrix has a pivot (leading 1) in RREF. There are no free variables. The final column of the RREF augmented matrix gives the exact, single value for each variable. Geometrically, this represents planes intersecting at a single point.
2. Consistent System with Infinitely Many Solutions
This occurs when the system has no contradictions, but there is at least one column in the coefficient matrix that does not contain a pivot. The variables corresponding to these columns are free variables. We express the pivot variables in terms of the free variables, yielding a parameterized general solution. Geometrically, this represents planes intersecting along a line or a higher-dimensional flat space.
3. Inconsistent System with No Solution
This occurs when the row reduction process creates a row with zeros on the left-hand side of the augmented divider and a non-zero value on the right-hand side (e.g., 0 = 1). Since this algebraic statement is false under all conditions, there is no set of variables that can satisfy all equations simultaneously. Geometrically, this represents parallel planes or planes that do not share a common intersection.