Ordered triple

Definition

An ordered triple is a set of three numbers written in a specific order, usually representing the solution to a system of three linear equations. It is generally denoted as $(x, y, z)$.

5 Must Know Facts For Your Next Test

1. An ordered triple represents coordinates in three-dimensional space.
2. It is used to find the point where three planes intersect.
3. Solving for an ordered triple often involves methods like substitution, elimination, or matrix operations such as Gaussian elimination.
4. Inconsistent systems have no solution, meaning no ordered triple satisfies all equations simultaneously.
5. Dependent systems have infinitely many solutions lying on a line or plane where all equations are satisfied.

Review Questions

• What does an ordered triple represent in a system of three linear equations?
• How can you determine if an ordered triple is the solution to a given system of equations?
• What methods can you use to solve for an ordered triple?

Related terms

System of Linear Equations: A collection of two or more linear equations involving the same set of variables.

Gaussian Elimination: A method used to solve systems of linear equations by transforming the system's augmented matrix into row-echelon form.

Three-Dimensional Coordinate System: A coordinate system that specifies each point uniquely with an ordered triple $(x, y, z)$.