Three Variables

Definition

Three variables refer to the presence of three unknown quantities or variables in a mathematical expression or equation. This concept is particularly relevant in the context of systems of linear equations, where the goal is to find the values of three unknown variables that satisfy a set of three linear equations.

5 Must Know Facts For Your Next Test

1. In a system of three linear equations with three variables, the goal is to find the unique values of the three unknown variables that satisfy all three equations simultaneously.
2. The system of three linear equations with three variables can be represented in matrix form as $Ax = b$, where $A$ is the coefficient matrix, $x$ is the vector of the three unknown variables, and $b$ is the vector of constants.
3. The augmented matrix, which combines the coefficient matrix $A$ and the constant vector $b$, can be used to solve the system of three linear equations using methods such as Gaussian elimination or Gauss-Jordan elimination.
4. The elimination method, which involves performing row operations on the augmented matrix to isolate one variable at a time, is a common technique for solving systems of three linear equations with three variables.
5. The solution to a system of three linear equations with three variables can be represented as an ordered triple $(x, y, z)$, where $x$, $y$, and $z$ are the unique values of the three unknown variables.

Review Questions

• Explain the purpose of a system of three linear equations with three variables.
  • The purpose of a system of three linear equations with three variables is to find the unique values of the three unknown variables that satisfy all three equations simultaneously. This system represents a set of constraints or relationships between the three variables, and solving the system allows for the determination of the specific values of the variables that meet all the given conditions.
• Describe how the augmented matrix can be used to solve a system of three linear equations with three variables.
  • The augmented matrix combines the coefficient matrix and the constant vector of a system of three linear equations with three variables, allowing for the use of matrix methods to solve the system. By performing row operations on the augmented matrix, such as Gaussian elimination or Gauss-Jordan elimination, the system can be transformed into a form where the values of the three unknown variables can be directly determined. The augmented matrix provides a structured and efficient way to solve the system of equations.
• Evaluate the role of the elimination method in solving systems of three linear equations with three variables.
  • The elimination method is a key technique for solving systems of three linear equations with three variables. By systematically performing row operations on the augmented matrix to isolate one variable at a time, the elimination method allows for the progressive elimination of variables until the unique values of all three unknown variables are found. This step-by-step process of eliminating variables is a crucial skill in solving systems of three linear equations, as it enables the determination of the specific solution that satisfies all the given constraints.

Related terms

System of Linear Equations: A set of linear equations that must be solved simultaneously to find the values of the unknown variables.

Augmented Matrix: A matrix that combines the coefficients of the variables and the constants from a system of linear equations, used to solve the system.

Elimination Method: A technique for solving a system of linear equations by systematically eliminating variables to find the values of the remaining variables.