A system of equations can have one solution, no solution, or infinite solutions.
Understanding System Solutions
When you have a system of equations, you are looking for values for the variables that satisfy all equations in the system simultaneously. For a system of linear equations (equations that graph as straight lines), there are precisely three possible outcomes regarding the number of solutions.
Based on the reference, "A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line)."
Let's break down these possibilities:
1. Single Solution
- This is the most common scenario.
- It means there is exactly one set of values for the variables that satisfies every equation in the system.
- Geometrically: For a system of two linear equations in two variables (like x and y), this occurs when the two lines intersect at a single point. That point's coordinates are the unique solution.
2. No Solution
- Also known as an inconsistent system.
- There are no values for the variables that can satisfy all equations at the same time.
- Geometrically: For two linear equations, this happens when the lines are parallel and distinct. Since they never intersect, there is no point (no solution) that lies on both lines.
3. Infinite Solutions
- Also known as a dependent system.
- Every solution to one equation is also a solution to the other equation(s).
- This happens when the equations are essentially the same, or one equation is a multiple of another.
- Geometrically: For two linear equations, this occurs when the lines are the same line (they coincide). Since every point on the line is common to both equations, there are infinitely many solutions.
Summary of Possible Solutions
Here is a quick overview of the possibilities for a system of linear equations:
| Number of Solutions | System Type | Geometric Interpretation (2D Linear) |
|---|---|---|
| One | Consistent | Lines intersect at one point |
| None | Inconsistent | Lines are parallel |
| Infinite | Dependent | Lines are the same |
Understanding these three cases is fundamental to solving and analyzing systems of equations.
