Unique solution

Definition

A unique solution in the context of systems of linear equations refers to a scenario where there is exactly one set of values for the variables that satisfies all equations in the system. This situation arises when the equations represent lines that intersect at a single point in a geometric representation, indicating that only one combination of variable values will simultaneously meet all given constraints.

5 Must Know Facts For Your Next Test

1. A system of linear equations has a unique solution if and only if the number of equations is equal to the number of variables and the equations are independent.
2. In graphical terms, a unique solution corresponds to two lines (in two dimensions) that intersect at exactly one point.
3. The determinant of the coefficient matrix must be non-zero for a linear system to have a unique solution, indicating that the lines are not parallel.
4. If a system has more equations than variables, it may still have a unique solution provided that no equation is redundant or contradictory.
5. Systems with unique solutions can often be solved using methods like substitution, elimination, or matrix operations such as row reduction.

Review Questions

• What conditions must be met for a system of linear equations to have a unique solution?
  • For a system of linear equations to have a unique solution, it must consist of as many independent equations as there are variables. This means that no equation can be derived from another, ensuring that they represent distinct lines or planes in a graphical context. Additionally, the determinant of the coefficient matrix should be non-zero, confirming that the lines are not parallel and will intersect at exactly one point.
• Compare and contrast systems with unique solutions versus dependent systems. How do their graphical representations differ?
  • Systems with unique solutions have exactly one intersection point when graphed, indicating a single set of variable values that satisfy all equations. In contrast, dependent systems yield infinitely many solutions because their equations represent the same line or plane, meaning they overlap completely on the graph. The distinction lies in their intersection behavior: unique solutions converge at one point while dependent systems coincide entirely.
• Evaluate how the concept of unique solutions impacts real-world applications such as economics and engineering.
  • The concept of unique solutions is vital in fields like economics and engineering where decision-making relies on precise outcomes. For instance, in optimizing resource allocation, having a unique solution ensures that there's a clear optimal strategy without ambiguity. In engineering design, it guarantees that specific parameters lead to functional designs without conflicting results. The implications extend to ensuring stability and predictability in complex systems where multiple variables interact under defined constraints.

Related terms

Consistent System: A consistent system is one in which at least one solution exists, which can be either unique or infinitely many.

Dependent System: A dependent system is a type of system of equations that has infinitely many solutions due to the equations representing the same line or plane.

Matrix Rank: The matrix rank is the dimension of the vector space generated by its rows or columns, and it determines the number of linearly independent rows or columns in a matrix, which is crucial for assessing the uniqueness of solutions.