Front

What is a system of three linear equations in three variables?

Back

A set of three equations that each represent a plane in three-dimensional space, where the variables are typically x, y, and z.

Front

What does it mean if a system of equations has no solution?

Back

It means that the planes represented by the equations do not intersect at any point.

Front

What does it mean if a system of equations has exactly one solution?

Back

It means that the three planes intersect at a single point.

Front

What does it mean if a system of equations has infinitely many solutions (IMS)?

Back

It means that the three planes intersect along a line or are coincident, meaning they lie on top of each other.

Front

What is the solution of a system of three linear equations in three variables called?

Back

Ordered Triple (x, y, z).

Front

How can you determine the number of solutions in a system of three equations?

Back

By analyzing the graphs of the equations to see how the planes intersect.

Front

What is the graphical representation of a system of three linear equations?

Back

It is represented by three planes in three-dimensional space.

Front

If two planes are parallel, what can be said about the system of equations?

Back

If two planes are parallel, the system has either no solution or infinitely many solutions, depending on the third plane.

Front

What is the significance of the intersection point of three planes?

Back

The intersection point represents the unique solution to the system of equations.

Front

What is the role of the third variable in a system of three equations?

Back

The third variable adds an additional dimension, allowing for more complex relationships between the equations.