Diagram 1 is the graph of the plane $$ 2x + 3y + z = 6$$ .
The red triangle is the portion of the plane when x, y, and z values are all positive. This plane actually continues off in the negative direction. All that is pictured is the part of the plane that is intersected by the positive axes (the negative axes have dashed lines).
Just like a system of linear equations with 2 variables is more than 1 line, a system of 3 variable equations is just more than 1 plane.
on Systems of 3 variable equations
Like systems of linear equations, the solution of a system of planes can be no solution, one solution or infinite solutions.
Below is a picture of three planes that have no solution. There is no single point at which all three planes intersect, therefore this system has no solution.
The other common example of systems of three variables equations that have no solution is pictured below. In the case below, each plane intersects the other two planes. However, there is no single point at which all three planes meet. Therefore, the system of 3 variable equations below has no solution.
of three variable systems
If the three planes intersect as pictured below then the three variable system has 1 point in common, and a single solution represented by the black point below.
of three variable systems
If the three planes intersect as pictured below then the three variable system has a line of intersection and therefore an infinite number of solutions.
