Unique Solution

Definition

A unique solution refers to a single set of values that satisfies a given system of equations. In the context of linear systems in two and three dimensions, it implies that there is exactly one point where the lines or planes intersect, making it the only combination of variable values that solves all equations simultaneously. This concept is crucial for determining the behavior of linear systems, especially when considering their graphical representations and algebraic formulations.

5 Must Know Facts For Your Next Test

1. A unique solution exists in a system of linear equations if the determinant of the coefficient matrix is non-zero.
2. For two linear equations in two dimensions, if the lines represented by those equations intersect at one point, that point is the unique solution.
3. In three dimensions, a unique solution corresponds to the intersection of three planes at a single point.
4. The conditions for a unique solution can also be determined using Gaussian elimination or matrix methods.
5. If a system has either no solutions or infinitely many solutions, then it does not have a unique solution.

Review Questions

• How can you determine if a system of linear equations has a unique solution?
  • To determine if a system of linear equations has a unique solution, check the determinant of the coefficient matrix. If the determinant is non-zero, it indicates that the equations are linearly independent and intersect at a single point. Additionally, methods such as Gaussian elimination can help solve the system and confirm the existence of that unique solution.
• What graphical representation can help visualize when a linear system has a unique solution?
  • Graphically, a unique solution in two dimensions is represented by two lines that intersect at exactly one point. In three dimensions, this can be visualized with three planes intersecting at a single point. If the lines or planes coincide or run parallel without intersection, it indicates either infinitely many solutions or no solutions, respectively.
• Evaluate how the concept of unique solutions applies to real-world problems involving optimization or resource allocation.
  • In real-world scenarios such as optimization problems or resource allocation, having a unique solution means that there is one best way to allocate resources or maximize efficiency given certain constraints. For example, in supply chain management, determining the optimal route for delivery trucks can be modeled as a linear system. A unique solution implies that there is one optimal route that minimizes costs or time without any ambiguity, ensuring that decisions based on this model lead to consistent and effective outcomes.

Related terms

Linear Independence: A property of a set of vectors in which no vector in the set can be expressed as a linear combination of the others.

Dimension: The number of coordinates needed to specify a point in a space; for example, a two-dimensional space requires two coordinates (x, y), while a three-dimensional space requires three (x, y, z).

Matrix Rank: The maximum number of linearly independent column or row vectors in a matrix, which indicates the dimension of the vector space generated by its rows or columns.