A linear equation can have either no solution, exactly one solution, or an infinite number of solutions. This depends on the specific equation and how it's structured.
Possible Solution Scenarios
The number of solutions for a linear equation depends on the relationship between the variables and constants involved. Let's explore the possibilities:
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No Solution: This occurs when the equation leads to a contradiction. For example, consider a system of two linear equations representing parallel lines. Because parallel lines never intersect, there's no solution that satisfies both equations simultaneously.
- Example:
- Equation 1: y = x + 1
- Equation 2: y = x + 2
These lines have the same slope but different y-intercepts, so they never intersect.
- Example:
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Exactly One Solution: This is the most common scenario. It happens when the equation can be solved for a unique value of the variable(s). In a system of two linear equations, this corresponds to the lines intersecting at a single point. That point represents the unique solution.
- Example:
- Equation 1: y = x + 1
- Equation 2: y = 2x - 1
These lines intersect at the point (2, 3), which is the unique solution.
- Example:
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Infinite Number of Solutions: This happens when the equation is an identity or when dealing with dependent systems of equations. In a system of two linear equations, this occurs when the lines are the same (coincident). Every point on the line is a solution.
- Example:
- Equation 1: y = x + 1
- Equation 2: 2y = 2x + 2 (which simplifies to y = x + 1)
These equations represent the same line, so any point on the line is a solution.
- Example:
Visual Representation (Two Linear Equations)
The reference illustrates this concept well with two lines:
| Scenario | Description | Number of Solutions |
|---|---|---|
| Intersecting Lines | The lines cross each other at one point. | Exactly One |
| Parallel Lines | The lines never intersect. | No Solution |
| Coincident (Same) Lines | The lines are on top of each other (identical line). | Infinite |
