Question
What does it mean for a system to have one solution?
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Answers
GPT-4o mini
In the context of a system of equations, saying that a system has "one solution" means that there is exactly one set of values for the variables that satisfies all the equations in the system simultaneously. This situation typically occurs in a linear system when the equations represent lines (in two dimensions) or planes (in three dimensions) that intersect at a single point.
Here are a few key points to consider:
1. **Unique Intersection**: In a graphical representation, one solution implies that the lines or planes intersect at exactly one point. This point corresponds to the values of the variables that make all the equations true.
2. **Linear Independence**: In a linear system, the equations must be linearly independent, meaning that they cannot be written as a linear combination of one another. This ensures that the solution is unique.
3. **Matrix Rank**: From a matrix perspective, if a system of equations can be represented as an augmented matrix, having one solution means that the rank of the coefficient matrix is equal to the rank of the augmented matrix, and both ranks equal the number of variables.
4. **Conditions for a Unique Solution**: For a system of linear equations to have a unique solution, the number of equations must be equal to the number of variables, and the equations must be consistent and independent.
In summary, a system having one solution indicates that the variables can be uniquely determined, allowing for a precise answer that satisfies all given conditions.
Here are a few key points to consider:
1. **Unique Intersection**: In a graphical representation, one solution implies that the lines or planes intersect at exactly one point. This point corresponds to the values of the variables that make all the equations true.
2. **Linear Independence**: In a linear system, the equations must be linearly independent, meaning that they cannot be written as a linear combination of one another. This ensures that the solution is unique.
3. **Matrix Rank**: From a matrix perspective, if a system of equations can be represented as an augmented matrix, having one solution means that the rank of the coefficient matrix is equal to the rank of the augmented matrix, and both ranks equal the number of variables.
4. **Conditions for a Unique Solution**: For a system of linear equations to have a unique solution, the number of equations must be equal to the number of variables, and the equations must be consistent and independent.
In summary, a system having one solution indicates that the variables can be uniquely determined, allowing for a precise answer that satisfies all given conditions.