A plane equation is a linear relation whose solutions form a plane in three-dimensional space. Written as ax + by + cz - d = 0, it assigns a z-value to each combination of x and y, covering the x-y plane with a surface. section 2.3.1 section 3.1.1
Use plane equations to connect algebra with geometry:
- Rewrite ax + by + cz = d as ax + by + cz - d = 0. section 3.1.1
- Interpret a system of three-variable linear equations as several planes; their common intersection is the solution. section 3.1.1
- Remember that two nonparallel planes intersect in a straight line, while parallel or coinciding planes do not determine one ordinary intersection line. section 2.3.1
A plane in explicit and general form
The relation z = 0.3x + 0.2y + 3 describes a plane by producing z for every x and y. Moving all terms to one side gives 0.3x + 0.2y - z + 3 = 0. section 2.3.1
Visualize how an explicit z-function becomes a plane and how the same surface appears in general form. section 2.3.1
