Gaussian Elimination
Reduce to row echelon form then back-substitute.
Key idea
Reduce to row echelon form then back-substitute.
In depth
Use row operations to reach row echelon form (staircase of pivots with zeros below), then back-substitute from the bottom up. It is the systematic way to solve linear systems.
Worked example
x + y = 3, 2x − y = 0: R₂ − 2R₁ gives −3y = −6, y = 2, then x = 1.
Real-world uses
- Solving large systems in structural analysis, traffic flow and economic input-output models
Common mistake
Making arithmetic errors when clearing below pivots; check the final answer.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Gaussian Elimination?
Reduce to row echelon form then back-substitute. It is a topic at the advanced level in Linear Algebra.
Where is Gaussian Elimination used in real life?
Real-world examples: Solving large systems in structural analysis, traffic flow and economic input-output models.
What should I learn before Gaussian Elimination?
Make sure you are comfortable with Row Operations, the previous topic in Linear Algebra.
What is a common mistake with Gaussian Elimination?
Making arithmetic errors when clearing below pivots; check the final answer.
How can I practise Gaussian Elimination?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Gaussian Elimination?
Move on to Gauss–Jordan Elimination next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- Projection – Component of one vector along another.
- Cross Product – A vector perpendicular to two vectors in 3D.
- Lines and Planes in Vector Form – Parametric and normal equations.