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Before diving into three variable systems of equations let's take a moment to review some of the fundamental concepts when graphing linear equations. Here we will review plotting points in two and three dimensions, graphing linear equations, and recall the meaning of a solution point.
Plotting Points - 2D vs 3D
The animations below illustrate the way we plot points with two coordinates, called an ordered pair, and with three coordinates called an ordered triple. Ordered pairs consist of x and y coordinates (x , y). Ordered triples consist of x, y, and z coordinates (x , y , z). In both cases, start at the origin and move according to each axes.
2 Variable Linear Equations VS 3 Variable Linear Equations
Standard Form of a Line
Ax + By = C
Need 2 points to write the equation of a line
Standard Form of a Plane
Ax + By + Cz = D
Need 3 points to write the equation of a plane
Solutions to Linear Equations & Systems
The flash cards to the right reviews the meaning of a solution point to a linear equation and to a linear system of equations. These are the fundamental concepts that will guide our understanding of solutions to three variable linear systems.
Solutions to Three Variable Systems
The top three images shown to the left represent a three variable system of linear equations with no solution. Recall that a solution is a point that exists on all three graphs together. In these images you only see intersections of two planes.
The bottom two images are examples with solutions. Bottom left shows a single point that all three planes have in common, hence one solution. Bottom right shows a line of intersection that all three planes have in common, hence infinite solutions becuase a line has infinte points on it.
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