Consistency and Solution Types
Unique, infinite or no solutions.
Key idea
Unique, infinite or no solutions.
In depth
A system has no solution if row reduction gives 0 = c (c ≠ 0), exactly one if every variable column has a pivot, and infinitely many if there are free variables. Compare rank of A and [A | b].
Worked example
A row [0 0 | 5] means no solution. A system with a free variable has infinitely many.
Real-world uses
- Checking whether a planned budget has a valid solution before committing to it
Common mistake
Assuming a system with more unknowns than equations must have a solution; it might not.
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 Consistency and Solution Types?
Unique, infinite or no solutions. It is a topic at the advanced level in Linear Algebra.
Where is Consistency and Solution Types used in real life?
Real-world examples: Checking whether a planned budget has a valid solution before committing to it.
What should I learn before Consistency and Solution Types?
Make sure you are comfortable with Gauss–Jordan Elimination, the previous topic in Linear Algebra.
What is a common mistake with Consistency and Solution Types?
Assuming a system with more unknowns than equations must have a solution; it might not.
How can I practise Consistency and Solution Types?
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 Consistency and Solution Types?
Move on to Homogeneous Systems 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.