A unique solution refers to a system of linear equations that has exactly one solution that satisfies all the equations in the system. This means that there is a single set of values for the variables that simultaneously solve all the equations, and no other set of values can do so.
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A unique solution can only occur in a consistent system of linear equations, where the equations do not contradict each other.
The number of variables in the system of linear equations must be equal to the number of equations for a unique solution to exist.
Graphically, a unique solution corresponds to a single point of intersection of the lines or planes representing the equations in the system.
Algebraically, a unique solution can be determined using methods such as substitution, elimination, or Gaussian elimination.
The existence of a unique solution is a key property in the study of systems of linear equations and their applications in various fields, including physics, engineering, and economics.
Review Questions
Explain the conditions necessary for a system of linear equations to have a unique solution.
For a system of linear equations to have a unique solution, the system must be consistent (the equations do not contradict each other) and the number of variables must be equal to the number of equations. This ensures that there is a single set of values for the variables that satisfies all the equations in the system. Graphically, a unique solution corresponds to a single point of intersection of the lines or planes representing the equations.
Describe how the method of Gaussian elimination can be used to determine the existence and uniqueness of a solution for a system of linear equations.
Gaussian elimination is a systematic method for solving systems of linear equations. By performing row operations on the augmented matrix of the system, Gaussian elimination can transform the system into an equivalent one that is easier to solve. If the final matrix has a non-zero diagonal, then the system has a unique solution. However, if the final matrix has a row of all zeros, then the system is either inconsistent (no solution) or has infinitely many solutions (not a unique solution).
Analyze the relationship between the number of variables and the number of equations in a system of linear equations and its impact on the existence of a unique solution.
For a system of linear equations to have a unique solution, the number of variables must be equal to the number of equations. If there are fewer equations than variables, the system will have infinitely many solutions, as there are more degrees of freedom. Conversely, if there are more equations than variables, the system will either have no solution (an inconsistent system) or a unique solution, depending on whether the equations are linearly independent. The balance between the number of variables and equations is a crucial factor in determining the existence and uniqueness of a solution for a system of linear equations.
A system of linear equations is a collection of two or more linear equations that share the same variables and must be solved together to find the values of those variables.