LINEAR ALGEBRA GUIDE
How Gaussian elimination solves linear systems
Gaussian elimination uses valid row operations to simplify an augmented matrix until each unknown can be found in a clear order.FORMULA
Allowed row operations: swap rows; multiply a row by a non-zero constant; add a multiple of one row to another rowWORKED EXAMPLE
For 2x + y = 8 and x − y = 1, row operations reduce the system until one equation gives y and the other gives x.STEP BY STEP
Write the system as coefficients
Place each equation in the same variable order and include the right-side values as the final augmented column.
Choose a pivot
Use a non-zero leading entry in the first column, then eliminate matching entries below it with row operations.
Continue by columns
Move to the next column and repeat until the matrix is in upper-triangular form or row-reduced echelon form.
Back-substitute or read the answer
Gaussian elimination uses back-substitution; Gauss-Jordan RREF reduces further so the solution can be read directly.