To determine the number of solutions for a system of three linear equations in three variables that is consistent and independent, we must first understand what these terms imply.
Consistent System: A system of equations is said to be consistent if it has at least one solution.
Independent Equations: In the context of a system of linear equations, "independent" means that the equations do not overlap in terms of redundancy (i.e., none of the equations are linear combinations of the others).
When a system of three linear equations in three variables is both consistent and independent, it implies the following:
1. The equations are not duplicates or multiples of each other.
2. The system does not lead to contradictions.
3. The variables have specific values that satisfy all three equations simultaneously.
For a consistent and independent system of three linear equations in three variables, all these conditions ensure that there is a unique intersection point of all three planes represented by these equations.
Therefore, such a system has exactly one solution. This means that the three planes intersect at a single point in the three-dimensional space, representing the unique solution to the system.
Given the choices:
- none
- one
- three
- infinitely many
The correct answer is "one". Hence, the number of solutions to this system is exactly one.
