Unique solution

Definition

A unique solution in the context of systems of linear equations refers to a specific outcome where there is exactly one pair of values that satisfies both equations simultaneously. This means the two lines represented by the equations intersect at a single point on the graph, which indicates that there is only one solution that makes both equations true. Understanding this concept is crucial as it distinguishes between various outcomes such as no solutions or infinitely many solutions, which are essential to grasp when analyzing linear systems.

5 Must Know Facts For Your Next Test

1. For two linear equations to have a unique solution, they must have different slopes, ensuring they intersect at one point.
2. The unique solution can be found using methods such as substitution, elimination, or graphing.
3. In a graphical representation, a unique solution corresponds to the intersection point of two non-parallel lines.
4. A system with a unique solution can be expressed in matrix form and solved using techniques like Gaussian elimination.
5. Identifying a unique solution is critical in real-world applications, where specific conditions need to be met based on the relationships described by the equations.

Review Questions

• How do you determine if a system of linear equations has a unique solution?
  • To determine if a system has a unique solution, you need to analyze the slopes of the lines represented by the equations. If the slopes are different, this indicates that the lines are not parallel and will intersect at exactly one point. This intersection point represents the unique solution. You can also use algebraic methods such as substitution or elimination to find this unique set of values.
• What role does the graphical representation play in understanding unique solutions for systems of equations?
  • The graphical representation is crucial for visualizing how linear equations interact with each other. A unique solution is illustrated by the intersection of two lines on a coordinate plane. When you see two lines crossing at one point, it confirms that there is exactly one solution that satisfies both equations. This visual aspect helps solidify your understanding of how systems work and aids in identifying different types of solutions.
• Evaluate how the concept of unique solutions applies in real-world situations involving linear systems.
  • In real-world situations, unique solutions often represent specific outcomes necessary for decision-making processes, such as determining costs or quantities in business contexts. For example, if two companies are analyzing their production costs and profits through linear equations, finding a unique solution allows them to pinpoint an exact price point where their products will break even or be profitable. Understanding when unique solutions occur can significantly impact strategic planning and resource allocation.

Related terms

Linear Equations: Equations that represent straight lines when graphed, typically in the form $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.

Graphical Intersection: The point at which two graphs meet, which represents the solution to a system of equations.

Dependent and Independent Systems: Dependent systems have infinitely many solutions, while independent systems have a unique solution, and inconsistent systems have no solutions.