Assume that the graph of a linear equation in three...

Assume that the graph of a linear equation in three variables is a plane in 3-dimensional space. (You will study these in Chapter 8.) (a) Explain geometrically how such a system can have a unique solution. (b) Explain geometrically how such a system can have no solution. Describe several possibilities. (c) Explain geometrically how such a system can have infinitely many solutions. Describe several possibilities. Construct physical models if you find that helpful.

Assume that the graph of a linear equation in three variables is a plane in 3-dimensional space. (You will study these in Chapter 8.)
(a) Explain geometrically how such a system can have a unique solution.
(b) Explain geometrically how such a system can have no solution. Describe several possibilities.
(c) Explain geometrically how such a system can have infinitely many solutions. Describe several possibilities. Construct physical models if you find that helpful.

Key Concepts

Plane in Three-Dimensional Space

A plane in three-dimensional space is a flat, two-dimensional surface that can be defined by a single linear equation in three variables. This concept is fundamental in analytic geometry and serves as the building block for understanding the spatial relationships between various linear equations in R³.

Intersection of Planes

When multiple planes are considered in three-dimensional space, their intersections can take various forms such as a line, a point, or even a whole plane. The nature of these intersections is key to understanding the solution set of a system of linear equations since each point of intersection represents a possible solution that satisfies all the equations simultaneously.

Unique Solution

A unique solution occurs when the intersection of all the planes is exactly one point. Geometrically, this typically happens when the planes intersect in a non-parallel, non-overlapping manner such that they cut each other transversely, resulting in a single common point in space that satisfies every equation in the system.

No Solution

A system has no solution when there is no point in common to all the intersecting planes. This inconsistency can arise in several scenarios, such as when the planes are parallel and never meet, or when two planes intersect in a line that is not intersected by the third plane. These configurations lead to a situation where no single point satisfies all the equations simultaneously.

Infinite Solutions

Infinite solutions occur when the intersections of the planes result in more than one common point, such as a line or a whole plane shared by all the equations. This situation typically arises when one of the planes is redundant (for example, it is exactly the same as another) or when the planes intersect exactly along a common line. In both cases, the system is underdetermined and has infinitely many solutions.

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