3.7. SYSTEMS OF EQUATIONS, GEOMETRY 53

The points, (x,y,z) satisfying an equation in three variables like 2x+4y−5z = 8 form aplane 1 and geometrically, when you solve systems of equations involving three variables,you are taking intersections of planes.

Consider the following picture involving two planes. Notice how these two planes

intersect in

New Plane

a line. It could also happen the two planes could fail to intersect.Now imagine a third plane. One thing that could happen is this third plane could have

an intersection with one of the first planes which results in a line which fails to intersect thefirst line as illustrated in the following picture.

Thus there is no point which lies in all three planes. The picture illustrates the situationin which the line of intersection of the new plane with one of the original planes forms aline parallel to the line of intersection of the first two planes. However, in three dimensions,it is possible for two lines to fail to intersect even though they are not parallel. Such linesare called skew lines. You might consider whether there exist two skew lines, each ofwhich is the intersection of a pair of planes selected from a set of exactly three planes suchthat there is no point of intersection between the three planes. You can also see that if youtilt one of the planes you could obtain every pair of planes having a nonempty intersectionin a line and yet there may be no point in the intersection of all three.

It could happen also that the three planes could intersect in a single point as shown inthe following picture.

New Plane

In this case, the three planes have a single point of in-tersection. The three planes could also intersect in a line.Thus in the case of three equations having three variables,the planes determined by these equations could intersect in asingle point, a line, or even fail to intersect at all. You seethat in three dimensions there are many possibilities. If youwant to waste some time, you can try to imagine all the thingswhich could happen but this will not help for more variablesthan 3 which is where many of the important applications lie.

Relations like x+y−2z+4w = 8 are often called hyper-planes.2 However, it is impossible to draw pictures of suchthings. The only rational and useful way to deal with thissubject is through the use of algebra not art. Mathematicsexists partly to free us from having to always draw pictures

in order to draw conclusions. The next chapter gives useful procedures which do not dependon pictures for finding solutions to systems of equations.

1Don’t worry about why this is at this time. It is not important. The discussion is intended to show you thatgeometric considerations like this don’t take you anywhere. It is the algebraic procedures which are importantand lead to important applications.

2The evocative semi word, “hyper” conveys absolutely no meaning but is traditional usage which makes theterminology sound more impressive than something like long wide flat thing.Later we will discuss some termswhich are not just evocative but yield real understanding.