A dependent system is a set of linear equations or functions where the variables are interdependent, meaning the value of one variable depends on the values of the other variables in the system. This concept is crucial in understanding the behavior and solutions of linear functions and systems of linear equations.
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In a dependent system, the variables are related, and the value of one variable can be expressed in terms of the other variables.
Dependent systems often arise when modeling real-world situations, such as in the context of linear functions and systems of linear equations.
The solution set of a dependent system is typically a line, plane, or hyperplane in the corresponding dimensional space.
Dependent systems can be either consistent, meaning they have at least one solution, or inconsistent, meaning they have no solution.
The existence and nature of the solution set for a dependent system depend on the specific coefficients and constants in the system of equations.
Review Questions
Explain how a dependent system of linear equations differs from an independent system.
In a dependent system of linear equations, the variables are interdependent, meaning the value of one variable depends on the values of the other variables in the system. This is in contrast to an independent system, where the variables are independent, and the value of one variable does not depend on the values of the other variables. The solution set of a dependent system is typically a line, plane, or hyperplane in the corresponding dimensional space, while the solution set of an independent system is a single point.
Describe the possible solution sets for a dependent system of linear equations.
The solution set of a dependent system of linear equations can be either consistent or inconsistent. A consistent dependent system has at least one solution that satisfies all the equations in the system, and the solution set is typically a line, plane, or hyperplane in the corresponding dimensional space. An inconsistent dependent system has no solution that satisfies all the equations in the system, meaning the system has no solution. The existence and nature of the solution set for a dependent system depend on the specific coefficients and constants in the system of equations.
Analyze how the concept of a dependent system relates to the topics of 2.2 Graphs of Linear Functions and 9.2 Systems of Linear Equations: Three Variables.
In the context of 2.2 Graphs of Linear Functions, a dependent system arises when the graphs of two or more linear functions intersect, indicating that the variables in the system are interdependent. The solution set of the dependent system corresponds to the point(s) of intersection of the linear function graphs. In the context of 9.2 Systems of Linear Equations: Three Variables, a dependent system occurs when the planes representing the equations in the system intersect along a line or plane, rather than at a single point. The solution set of the dependent system is the line or plane of intersection, which represents the set of points that satisfy all the equations in the system.
Related terms
Independent System: An independent system is a set of linear equations or functions where the variables are independent, meaning the value of one variable does not depend on the values of the other variables in the system.