The goal is a staircase of zeros below the diagonal. Once the matrix has that shape, the last equation has one unknown and the rest unravel.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Pivots
A pivot is the first non-zero entry in a row. Each pivot is used to clear everything beneath it.
Clearing a column
Subtract the right multiple of the pivot row from each row below. The chosen multiple is whatever makes that entry zero.
Back substitution
Solve the bottom equation, substitute upward, and repeat. Each step has exactly one new unknown.
Going further
Clearing above the pivots too, and scaling each pivot to 1, gives reduced row echelon form. The answers then sit in the last column.
Why it is the standard method
Elimination is systematic and never gets stuck. It is what a computer does, on systems with thousands of unknowns.
Clear entries until the answer is visible
Elimination drives the matrix towards row echelon form, where each row starts further right than the one above. In that form the last equation involves one unknown and back-substitution finishes it.
Pivots are the leading entries
The first nonzero entry in each row is a pivot, and its column is a pivot column. Counting pivots is how the number of solutions and the rank are read off, so identifying them matters.
Reduced row echelon form
Continuing until each pivot is 1 with zeros above and below gives the reduced form, in which the solution can be read directly with no back-substitution. It is unique for a given matrix.
It is a genuine algorithm
Elimination terminates in a predictable number of steps regardless of the matrix. That is why it is what computers actually run, and why solving large systems is a routine rather than an art.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Sofia Eliminate
Sofia reduces the rows [1, 2 | 7] and [3, 1 | 11].
- Step 1
She uses the leading 1 in row 1 as the pivot.
Step 4: Your Turn
Practice makes it stick.
The Bottom Row
Problem 1 of 2
A reduced row reads [0, 4 | 12]. What is y?
The Substitution
Problem 2 of 2
Row 1 reads [1, 2 | 7] and you know y = 2. What is x?
Reduce and Solve
1 of 8
A row reads [0, 3 | 9]. What is y?
2 of 8
A row reads [0, −2 | 8]. What is y?
3 of 8
Row 1 is [1, 1 | 10] and y = 4. What is x?
4 of 8
Clearing the entry 5 below a pivot of 1. What multiple of the pivot row is subtracted?
5 of 8
In reduced row echelon form, what is every pivot scaled to?
6 of 8
A 3 by 3 system fully reduced. How many pivots does it have if it has a unique solution?
7 of 8
Order the steps of Gaussian elimination.
- 1Clear every entry beneath that pivot
- 2Move to the next column and repeat
- 3Back-substitute from the bottom row upward
- 4Choose a pivot in the first column
8 of 8
A row reads [0, 1 | −6]. What is y?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A row reads [0, 5 | 20]. What is y?
Question 2 of 2
What shape is Gaussian elimination aiming for?
What You Learned
- Elimination clears entries below each pivot until the matrix is a staircase.
- The bottom row then has one unknown, and back substitution unravels the rest.
- Reduced row echelon form goes further and leaves the answers in the last column.