key term - Free Variable

Definition

A free variable is a variable in a mathematical expression or equation that can take on any value without affecting the validity of the expression. It is a variable that is not constrained or determined by any other variables or conditions within the context of the problem.

5 Must Know Facts For Your Next Test

1. In the context of solving systems of equations using Gaussian elimination, the free variables represent the variables that are not determined by the system of equations.
2. The number of free variables in a system of equations is equal to the difference between the number of variables and the number of linearly independent equations in the system.
3. Free variables play a crucial role in the solution of a system of equations, as they allow for the expression of the solution in terms of one or more parameters.
4. The presence of free variables in a system of equations indicates that the system has infinitely many solutions, with the free variables acting as the parameters that can be assigned different values to generate the various solutions.
5. Identifying and understanding the role of free variables is essential in interpreting the solutions of systems of equations and understanding the properties of the solution set.

Review Questions

• Explain the relationship between the number of free variables and the number of linearly independent equations in a system of equations.
  • The number of free variables in a system of equations is equal to the difference between the number of variables and the number of linearly independent equations in the system. This relationship is important because it determines the number of parameters that can be used to express the solution of the system. If there are more variables than linearly independent equations, the system will have free variables, indicating that the solution set is infinite and can be expressed in terms of the free variables.
• Describe how the presence of free variables affects the solution set of a system of equations.
  • The presence of free variables in a system of equations indicates that the system has infinitely many solutions. The free variables act as parameters that can be assigned different values to generate the various solutions. This means that the solution set is not a single point, but rather a set of points or a subspace within the space defined by the variables. Understanding the role of free variables is crucial in interpreting the solutions of systems of equations and understanding the properties of the solution set.
• Analyze the importance of identifying and understanding free variables in the context of solving systems of equations using Gaussian elimination.
  • Identifying and understanding the role of free variables is essential when solving systems of equations using Gaussian elimination. The free variables represent the variables that are not determined by the system of equations, and they play a crucial role in the final solution. By recognizing the free variables, you can express the solution in terms of one or more parameters, which provides a more complete and informative representation of the solution set. This understanding is vital for interpreting the solutions, understanding the properties of the solution set, and making informed decisions based on the system of equations.

Related terms

Dependent Variable:

A dependent variable is a variable whose value depends on the value of one or more independent variables in a mathematical relationship or equation.

Independent Variable:

An independent variable is a variable that is manipulated or changed to observe the effect on a dependent variable in a mathematical relationship or equation.

System of Equations:

A system of equations is a set of two or more equations that share one or more variables, and the solution to the system is the set of values for the variables that satisfy all the equations simultaneously.