Things get more interesting with two-by-three linear systems (two equations and three
variables x,y, and z). The graph appears as two planes in Cartesian three-space. The planes
might intersect in a straight line, in which case there are infinitely many solutions: all the
ordered triples (x,y,z) that correspond to points on that line. A second possibility is that the
two planes are parallel. Then they don’t intersect anywhere, and there are no solutions to
the system. A third possibility is that the planes coincide. Then, again, there are infinitely
many solutions: all the ordered triples (x,y,z) that correspond to points on that plane. Two flat
planes in space can never intersect in a single point. That means a two-by-three linear system
can’t have a unique solution.
More equations than variables
When a linear system in two variables has more than two equations, or a linear system in
three variables has more than three equations, we have “extra data.” We can imagine such
a system as a two-by-two or three-by-three linear system with one or more extra equations
thrown in.
If a two-by-two or three-by-three linear system is inconsistent, then adding one or more
extra equations cannot make it consistent. If a two-by-two or three-by-three linear system is
consistent, then adding one or more extra equations might make it inconsistent or redundant,
but not necessarily.
These statements can be generalized to any number of linear equations in any number
of variables. If we encounter 27 equations in 28 variables, we can be certain that the system
has no unique solution. But if we see 28 or more equations in 28 variables, we can’t know if
the system has a unique solution until we try to solve it, preferably with the help of a com-
puter. If the system has no unique solution, the algebra will lead us into absurd or useless
statements.
Here’s a challenge!
Draw a graph of the following three-by-two linear system to illustrate why it does not have a unique solution:
y=x+ 1
y= 3 x− 1
y=− 2 x− 3
Solution
These three equations are in SI form. The y-intercepts are 1, −1, and −3, respectively. The slopes are 1,
3, and −2 respectively; they can be used to find second points as shown in Fig. 18-3, which portrays the
graphs of the equations as a single system. Each line intersects both of the others, but there is no single
point common to all three lines. For any linear system to have a solution, no matter how many variables
there are, the graphs of all the equations must have a single point in common. If there are two variables,
that point can be represented by a unique ordered pair. If there are three variables, the point can be repre-
sented by a unique ordered triple. If there are n variables, the point can be represented by a unique ordered
n-tuple.
General Linear Systems 293