Objectives

1. Learn to replace a system of linear equations by an augmented matrix.
   
2. Learn how the elimination method corresponds to performing row operations on an augmented matrix.
   
3. Understand when a matrix is in (reduced) row echelon form.
   
4. Learn which row reduced matrices come from inconsistent linear systems.
   
5. Recipe: the row reduction algorithm.
   
6. Vocabulary words: row operation, row equivalence, matrix, augmented matrix, pivot, (reduced) row echelon form.
   

In this section, we will present an algorithm for “solving” a system of linear equations.

We will solve systems of linear equations algebraically using the elimination method. In other words, we will combine the equations in various ways to try to eliminate as many variables as possible from each equation. There are three valid operations we can perform on our system of equations:

• Scaling: we can multiply both sides of an equation by a nonzero number.
  

  Cx+2y+3z=62x−3y+2z=143x+y−z=−2multiply1stby−3−−−−−−−−−→C−3x−6y−9z=−182x−3y+2z=143x+y−z=−2
  

• Replacement: we can add a multiple of one equation to another, replacing the second equation with the result.
  

  Cx+2y+3z=62x−3y+2z=143x+y−z=−22nd=2nd−2×1st−−−−−−−−−−→Cx+2y+3z=6−7y−4z=23x+y−z=−2
  

• Swap: we can swap two equations.
  

  Cx+2y+3z=62x−3y+2z=143x+y−z=−23rd←→1st−−−−−−→C3x+y−z=−22x−3y+2z=14x+2y+3z=6