A unique solution refers to a single, specific answer that satisfies the given system of equations or linear equation. It is the only solution that makes all the equations in the system true simultaneously.
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A unique solution means there is only one set of values for the variables that satisfies all the equations in the system.
For a system of linear equations to have a unique solution, the system must be consistent and the equations must be linearly independent.
In a consistent system with a unique solution, the graphs of the linear equations intersect at a single point.
Solving a system of linear equations using the substitution or elimination method can lead to a unique solution if the system is consistent and the equations are linearly independent.
Applications involving systems of equations, such as word problems, often require finding the unique solution that represents the answer to the real-world problem.
Review Questions
Explain the relationship between a unique solution and a consistent system of linear equations.
For a system of linear equations to have a unique solution, the system must be consistent, meaning that the equations have at least one common solution that satisfies all the equations. Additionally, the equations in the system must be linearly independent, which means that no equation can be expressed as a linear combination of the other equations. When these conditions are met, the system will have a single, unique solution that represents the only set of variable values that make all the equations true simultaneously.
Describe how the graphical representation of a system of linear equations relates to the existence of a unique solution.
When a system of linear equations has a unique solution, the graphs of the individual equations in the system will intersect at a single point. This point of intersection represents the unique solution, as it is the only set of variable values that satisfies all the equations in the system. If the graphs of the equations do not intersect at a single point, then the system either has no solution (parallel lines) or infinitely many solutions (coincident lines), and a unique solution does not exist.
Analyze the role of unique solutions in the context of solving applications with systems of equations.
In applications involving systems of equations, such as word problems, finding the unique solution is crucial because it represents the only set of variable values that accurately answers the real-world problem. The unique solution provides the specific information needed to solve the application, whether it's determining the number of items to purchase, the amount of money to invest, or the dimensions of a structure. Without a unique solution, the application cannot be fully resolved, as there would be multiple possible answers that satisfy the given constraints. Therefore, the ability to identify and interpret the unique solution is essential for successfully solving application problems involving systems of equations.