Gaussian Elimination and Row Echelon Form
Reduce a system to echelon form with three row operations that never change the solution set, then read pivots, free variables and the full solution off the result.
What a learner can do afterwards
- Carry a system to reduced row echelon form
- Identify pivot and free columns and write the general solution
- Say why each row operation preserves the solution set
1 · Read
Elimination clears unknowns one column at a time using only three moves. You may swap two rows, scale a row by a nonzero number, or add a multiple of one row to another. Each move is reversible, so the solution set never changes while the system gets simpler. Work top to bottom toward staircase form, then back substitute from the last pivot up.
Solve x plus y equals 5 with x minus y equals 1. Add the equations and y cancels: 2 x equals 6, so x equals 3. Then y equals 2, and both equations hold: 3 plus 2 is 5, and 3 minus 2 is 1. Adding was the allowed move of adding one row to another.
Staircase form exposes pivots and free columns at a glance. Pivot columns hold leading entries, while the rest are free and each gets a parameter in the general solution. Push to reduced form, with 1s as pivots and zeros above and below, and the solution reads itself off. A 3 by 4 system with pivots in columns 1 and 2 leaves columns 3 and 4 free.
Read the finished form like a report. A row reading 0 equals 5 announces no solution, which no values can satisfy. Count free variables as unknowns minus pivots: 5 unknowns with 3 pivots leave 2 free. Rectangular systems behave exactly like square ones, with the pivot count doing the talking.
Swap, scale, and add toward staircase form, then read pivots, free columns, and the solution.
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8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.