Dependent

Definition

In the context of solving systems of equations using matrices, a dependent variable is a variable whose value is determined by the values of other variables in the system. It represents an unknown quantity that can be expressed in terms of the other variables.

5 Must Know Facts For Your Next Test

1. The number of dependent variables in a system of equations is equal to the number of equations in the system.
2. The values of the dependent variables are determined by solving the system of equations using matrix methods, such as Gaussian elimination or the inverse of the coefficient matrix.
3. Dependent variables are often represented by the letters $x$, $y$, $z$, etc., while the independent variables are represented by the letters $a$, $b$, $c$, etc.
4. The rank of the augmented matrix of a system of equations determines the number of independent variables in the system, and the remaining variables are considered dependent.
5. Dependent variables can be expressed in terms of the independent variables using the solutions obtained from the matrix methods.

Review Questions

• Explain the role of dependent variables in a system of linear equations and how they are related to the independent variables.
  • In a system of linear equations, the dependent variables are the unknown quantities that can be expressed in terms of the independent variables. The values of the dependent variables are determined by solving the system of equations using matrix methods, such as Gaussian elimination or the inverse of the coefficient matrix. The number of dependent variables is equal to the number of equations in the system, and they are typically represented by the letters $x$, $y$, $z$, etc. The relationship between the dependent and independent variables is determined by the coefficients in the system of equations and the solutions obtained through the matrix methods.
• Describe how the rank of the augmented matrix of a system of equations is related to the number of independent and dependent variables in the system.
  • The rank of the augmented matrix of a system of linear equations determines the number of independent variables in the system. The remaining variables are considered dependent variables. The rank of the matrix is the number of linearly independent rows or columns, which corresponds to the number of independent variables. The number of dependent variables is then equal to the number of equations in the system, minus the number of independent variables. This relationship between the rank of the matrix and the number of independent and dependent variables is crucial in understanding and solving systems of equations using matrix methods.
• Explain how the solutions obtained from solving a system of equations using matrix methods can be used to express the dependent variables in terms of the independent variables.
  • When solving a system of linear equations using matrix methods, such as Gaussian elimination or the inverse of the coefficient matrix, the solutions provide the values of the dependent variables in terms of the independent variables. These solutions can be used to express the dependent variables as functions of the independent variables. This is an important step in understanding the relationships between the variables in the system and in interpreting the results of the matrix-based solution process. By expressing the dependent variables in terms of the independent variables, you can gain deeper insights into the structure and behavior of the system of equations.

Related terms

Independent Variable: A variable in a system of equations that can be assigned any value, and the values of the other variables depend on it.

Augmented Matrix: A matrix that combines the coefficients of the variables and the constants in a system of linear equations.

Rank of a Matrix: The number of linearly independent rows or columns in a matrix, which determines the number of independent variables in a system of equations.