key term - Free Variable

Definition

A free variable is a variable in a system of equations that can be assigned any value without affecting the validity of the solution. It is a variable that is not constrained by the equations in the system, allowing for flexibility in finding a solution.

5 Must Know Facts For Your Next Test

1. In a system of three linear equations with three variables, there is typically one free variable that can be assigned any value to find a solution.
2. The presence of a free variable in a system of linear equations means that there are infinitely many solutions, as the free variable can be assigned different values.
3. Gaussian elimination is a method used to solve systems of linear equations, and the presence of a free variable affects the steps and the final solution.
4. The free variable in a system of linear equations allows for flexibility in finding a solution, as it can be used to satisfy additional constraints or optimize a particular outcome.
5. Understanding the concept of free variables is crucial in solving systems of linear equations, as it helps in identifying the degrees of freedom and the structure of the solution set.

Review Questions

• Explain the role of a free variable in a system of three linear equations with three variables.
  • In a system of three linear equations with three variables, the presence of a free variable means that one of the variables can be assigned any value without affecting the validity of the solution. This free variable allows for flexibility in finding a solution, as it can be used to satisfy additional constraints or optimize a particular outcome. The other two variables are then determined based on the values of the free variable and the equations in the system.
• Describe how the concept of free variables affects the steps and the final solution when using Gaussian elimination to solve a system of linear equations.
  • When using Gaussian elimination to solve a system of linear equations, the presence of a free variable changes the steps and the final solution. Gaussian elimination involves transforming the system into an equivalent system with an upper triangular coefficient matrix. If a free variable is present, the final solution will contain this free variable, which can be assigned any value to find a solution. This means that the solution set will be a family of solutions, rather than a single unique solution. The steps of Gaussian elimination must be adjusted to account for the free variable and the flexibility it provides in finding a solution.
• Analyze the significance of understanding the concept of free variables in the context of solving systems of linear equations.
  • Understanding the concept of free variables is crucial in solving systems of linear equations, as it helps in identifying the degrees of freedom and the structure of the solution set. The presence of a free variable indicates that there are infinitely many solutions, as the free variable can be assigned different values. This knowledge allows you to approach the problem with a deeper understanding, enabling you to find the most appropriate solution based on additional constraints or optimization criteria. By recognizing and working with free variables, you can more effectively solve systems of linear equations and gain valuable insights into the underlying mathematical structure of the problem.

Related terms

Dependent Variable:

A dependent variable is a variable whose value depends on the values of the other variables in the system of equations.

Gaussian Elimination:

Gaussian elimination is a method used to solve systems of linear equations by transforming the system into an equivalent system with an upper triangular coefficient matrix.

System of Linear Equations:

A system of linear equations is a set of two or more linear equations with two or more variables, where the variables represent unknown quantities.