Glossary
System of equations
Also written: systems of equations · simultaneous equations
Definition
A system of equations is two or more equations that must all be true at once. For two lines, the solution is the point where their graphs cross.
Notation
A brace groups the equations that must hold together:
The solution is the pair satisfying both, here .
Why the solution is the intersection
A graph is the set of all solutions of its equation. So a point on the first line solves the first equation, a point on the second solves the second, and a point on both solves both at once — which is what the system asks for.
Two different straight lines meet at most once, so a system of two lines has at most one solution.
Three outcomes
| Lines | Solutions |
|---|---|
| cross once | exactly one |
| parallel | none — they never meet |
| identical | infinitely many — every point works |
When you solve algebraically and the variables all cancel, the leftover statement tells you which case you are in: something false like means parallel, and something true like means one line written twice.
Methods
Graphing shows what is happening. Substitution suits a system where a variable is already alone; elimination suits one where coefficients match or are opposites. See systems of equations.
Lessons that use this term
- Solving Systems of Linear Equations
Solve a system by graphing, substitution or elimination, why the solution is the intersection point, and how to spot systems with none or infinitely many.