No, infinite solutions mean a system of equations is consistent and dependent, not inconsistent.
Here's a breakdown:
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Consistent System: A system of equations is considered consistent if it has at least one solution (either one solution or an infinite number of solutions).
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Inconsistent System: A system of equations is considered inconsistent if it has no solution.
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Dependent System: A consistent system with an infinite number of solutions is considered dependent. This typically means the equations represent the same line or plane (in higher dimensions). One equation is essentially a multiple of the other.
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Independent System: A consistent system with exactly one solution is considered independent.
Here's a table summarizing the relationships:
| System Type | Number of Solutions | Description |
|---|---|---|
| Consistent | One solution | Independent. The equations intersect at a single point. |
| Consistent | Infinite solutions | Dependent. The equations represent the same line (or plane in higher dimensions). Every solution of one equation is also a solution of the other. |
| Inconsistent | No solution | The equations represent parallel lines (or planes that never intersect). There is no solution that satisfies all equations simultaneously. |
Example:
Consider the following system of equations:
x + y = 22x + 2y = 4
Notice that the second equation is simply the first equation multiplied by 2. Therefore, they represent the same line. Any solution to the first equation will also be a solution to the second equation. This system has infinite solutions (e.g., (0, 2), (1, 1), (2, 0), etc.). This is a consistent and dependent system.
On the other hand, the system:
x + y = 2x + y = 3
has no solution because there are no values of x and y that can simultaneously satisfy both equations. This is an inconsistent system.
In conclusion, if a system of equations has infinite solutions, it is classified as consistent (specifically, consistent and dependent), not inconsistent. Inconsistent systems have no solutions.
