Number of solutions
A linear system has exactly one solution, no solution, or infinitely many. During elimination, a row 0 = 5 means none, and a row of zeros means free variables.
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Analogy
Two lines can cross once, be parallel, or lie exactly on top of each other.
Example
x + y = 2 and 2x + 2y = 4:same line, infinitely many solutions.x + y = 2 and x + y = 3:parallel, no solution.
In practice
Tells you whether a model is consistent before you try to solve it.
Practice
Flashcards
- How many solutions can a linear system have?One, none, or infinitely many.
Interview questions
- Explain the three possible outcomes for the number of solutions to a linear system with an example of each.
- One solution: lines cross once
- None: parallel lines, a row 0 = non-zero
- Infinitely many: same line, a free variable
- Elimination reveals which case occurs
Questions people ask
How can I tell there is no solution?
During elimination you reach a row of zeros on the left with a non-zero number on the right, such as 0 = 1. That statement is impossible, so no choice of unknowns can satisfy all the equations. The system is called inconsistent.
What is a free variable?
When elimination leaves fewer useful equations than unknowns, some unknowns can take any value, and the others depend on them. For x plus y equals 2, y is free, and x equals 2 minus y. That gives infinitely many solutions.