290 Larger Linear Systems
General Linear Systems
Now that you’ve seen two-by-two and three-by-three linear systems, you might wonder,
“What does a three-by-three graph look like?” or “What happens when there are many linear
equations in many variables?” or “What happens when the number of equations is not the
same as the number of variables?”Two-by-two geometry
The graph of a linear equation in two variables shows up as a straight line in the Cartesian
plane, as you saw in Chap. 17. When you have two such equations, their graphs always appear
in one of three ways:- Two different lines that intersect at a single, unique point
- Two different lines that are parallel
- Two lines that precisely coincide
In the first case, the system has a single solution, corresponding to the point where the
lines intersect. In the second case, there is no solution. In the third situation, there are infi-
nitely many solutions.Cartesian three-space
The graph of a linear equation in three variables can’t be drawn on a Cartesian plane. Instead,
we need to use a system that can portray all of space. The most common such system is called
Cartesian three-space. It makes use of three coordinate axes, all of which intersect at their zero
points, and in such a way that each axis is perpendicular to the other two.
Cartesian three-space is sometimes drawn in perspective, as in Fig. 18-1. In this exam-
ple, the variables are x,y, and z. If the drawings were literal, the x axis would appear hori-
zontal on the page, the y axis would appear vertical on the page, and the z axis would be
perpendicular to the page. Note that the positive x axis goes to the right, the positive y axis
goes upward, and the positive z axis comes toward us.
Figure 18-2 shows two specific points, P and Q, plotted in Cartesian three-space. The
coordinates of point P are (−5,−4, 3), and the coordinates of point Q are (3, 5, −2). Points
are denoted as ordered triples in the form (x, y, z), where the first number represents the value
on the x axis, the second number represents the value on the y axis, and the third number
represents the value on the z axis.Three-by-three geometry
The graph of a linear equation in three variables appears as a flat plane (not a line!) in Carte-
sian three-space. When we have three linear equations in three variables, their graphs can show
up in any of the following ways:- Three different planes that all intersect at a single, unique point.
- Three different planes, two of which are parallel, and the third of which intersects the
other two in a pair of parallel lines.