A system of linear equations is represented by its augmented matrix.
High school manipulations of the equations correspond to row operations on the augmented matrix.
Maneuvers to put the matrix in the form where the system is (essentially) solved involve proceeding in a systematic way.
Vector addition x1x2…xn+y1y2…yn=x1+y1x2+y2…xn+yn
Multiplication of a vector by a scalar cx1x2…xn=cx1cx2…cxn
Replace a row by its sum with a multiple of another row.
Switch two rows.
Replace a row by a non-zero multiple of itself.
All non-zero rows precede all zero rows.
The leading non-zero elements in the non-zero rows are staggered.
It is in row echelon form.
The first non-zero element in a non-zero row is a 1.
A leading 1 is the only non-zero element in its column.
x1−2x2+x3=0 2x2−8x3=8 −4x1+5x2+9x3=−9
Augmented matrix:
x=29 y=16 z=3