key term - Ordered Triple

Definition

An ordered triple is a set of three numbers or values that are arranged in a specific order, typically denoted as (x, y, z). It is a fundamental concept in the context of solving systems of equations with three variables, as it represents a unique point in three-dimensional space.

5 Must Know Facts For Your Next Test

1. An ordered triple is a set of three numbers or values that are arranged in a specific order, typically denoted as (x, y, z).
2. Ordered triples are used to represent points in three-dimensional space, where the three coordinates correspond to the x, y, and z axes.
3. In the context of solving systems of equations with three variables, an ordered triple represents a solution that satisfies all the equations in the system.
4. The order of the numbers in an ordered triple is important, as (x, y, z) is not the same as (x, z, y) or (y, x, z).
5. Ordered triples can be used to represent various real-world phenomena, such as the location of an object in 3D space, the components of a vector, or the values of three related variables.

Review Questions

• Explain how an ordered triple is used to represent a point in three-dimensional space.
  • An ordered triple (x, y, z) represents a unique point in three-dimensional space, where the x-coordinate corresponds to the position along the x-axis, the y-coordinate corresponds to the position along the y-axis, and the z-coordinate corresponds to the position along the z-axis. This three-dimensional coordinate system allows for the precise specification of a location in 3D space, which is essential for solving systems of equations with three variables.
• Describe the relationship between an ordered triple and a solution to a system of equations with three variables.
  • When solving a system of equations with three variables, the solution is represented by an ordered triple (x, y, z) that satisfies all the equations in the system. Each value in the ordered triple corresponds to the value of a specific variable, and the ordered triple as a whole represents a point in three-dimensional space that is the solution to the system of equations. The order of the values in the ordered triple is crucial, as it determines which variable is associated with each coordinate.
• Analyze how the order of the values in an ordered triple affects its meaning and interpretation.
  • The order of the values in an ordered triple is essential, as (x, y, z) is not the same as (x, z, y) or (y, x, z). The order of the values determines which variable is associated with each coordinate, and this has a significant impact on the meaning and interpretation of the ordered triple. For example, in the context of a system of equations with three variables, the ordered triple (2, 3, 4) would represent a different solution than the ordered triples (2, 4, 3) or (3, 2, 4). The order of the values in an ordered triple must be carefully maintained to ensure the correct interpretation and application of the concept.

Related terms

System of Equations:

A system of equations is a set of two or more equations that must be solved simultaneously to find the values of the variables that satisfy all the equations.

Three-Dimensional Space: Three-dimensional space, also known as 3D space, is a geometric representation of the physical universe in which we live. It is defined by three perpendicular axes (x, y, and z), allowing the location of a point to be specified by three coordinates.

Cartesian Coordinate System: The Cartesian coordinate system is a way of specifying the location of a point in space using three numerical coordinates, x, y, and z, which represent the distances from the point to three mutually perpendicular planes.