key term - Unique Solution

Definition

A unique solution refers to a situation in a system of equations where exactly one set of values satisfies all equations simultaneously. This characteristic is essential for determining the intersection of lines in graphical representations or for solving linear equations, ensuring that there is a distinct point at which the lines meet. The concept highlights the importance of consistency and independence in systems of equations, indicating that there are no redundant or conflicting equations present.

5 Must Know Facts For Your Next Test

1. For a linear system to have a unique solution, the equations must represent lines that intersect at exactly one point in a two-dimensional space.
2. In matrix form, a unique solution exists when the coefficient matrix has full rank, meaning its rank equals the number of variables.
3. Graphically, a unique solution can be visually identified as the point where two lines cross, indicating that their slopes are different.
4. In contrast to a unique solution, if a system has no solutions, it is called inconsistent, while if it has infinitely many solutions, it is dependent.
5. Determining whether a system has a unique solution can often involve techniques such as substitution, elimination, or using determinants in matrix operations.

Review Questions

• How can you determine if a system of linear equations has a unique solution based on its graphical representation?
  • To determine if a system of linear equations has a unique solution graphically, you would plot each equation on the same coordinate plane. If the lines intersect at exactly one point, this indicates that there is one unique solution. If the lines are parallel and never meet, there is no solution, and if they coincide completely, there are infinitely many solutions. Therefore, distinct intersection points signify a unique solution.
• Discuss the implications of having a unique solution in terms of the consistency and independence of equations within a system.
  • Having a unique solution implies that the system of equations is consistent and independent. Consistency means that at least one solution exists, while independence indicates that no equation can be derived from another; thus, they provide distinct information. If any equation were dependent on another, it would lead to either no solutions or infinitely many solutions instead of a single unique outcome. Hence, uniqueness reinforces the reliability and distinctiveness of each equation in providing information about the variables.
• Evaluate how using matrices can help you identify whether a system of linear equations has a unique solution, considering matrix properties and operations.
  • Using matrices to identify whether a system has a unique solution involves examining the properties of the coefficient matrix. By calculating its rank and comparing it to the number of variables, you can conclude if a unique solution exists. Specifically, if the rank equals both the number of variables and the number of equations, this confirms that there is one unique solution. Additionally, employing operations such as row reduction can simplify matrices to reveal their independence or dependence among rows, further clarifying the structure of solutions.

Related terms

Consistent System:

A system of equations that has at least one solution, which may be unique or infinite.

Dependent System: A system of equations that has an infinite number of solutions because the equations represent the same line.

Matrix Rank: The maximum number of linearly independent row or column vectors in a matrix, which helps determine the number of solutions to a system of equations.