Frequently Asked Questions
How are linear systems solved?
By transforming the equations using elementary operations (swap, scale, add) until the unknowns are isolated. Numerically this is Gaussian elimination; on paper it is substitution or elimination.
When is a system inconsistent?
When the coefficient matrix is singular and the right-hand side does not lie in the column space. After elimination one row reads 0 equals a nonzero number.
Why prefer Gaussian elimination?
It runs in n cubed time, generalizes to any size, and with partial pivoting is numerically stable. Cramer's rule is slow for big n; direct inverse is unstable.
How do you solve a 3x3 system of equations?
Use Gaussian elimination: form the augmented matrix, use row operations to eliminate one variable at a time until the matrix is triangular, then back-substitute. Partial pivoting (swapping in the largest pivot) keeps the arithmetic stable.
What does it mean when a system has no solution?
It means the equations are inconsistent: elimination produces a false statement like 0 = 5, and the coefficient matrix is singular (determinant zero). Geometrically the lines or planes never meet at a common point.
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