SEC. 7.3 Linear Systems of Equations. Gauss E[imination 289
Geometric lnterpretation. Existence and Uniqueness of Solutions
If m : ft : 2, we have two equátions in two unknowns x1, x2
a1 1lal2x2:b1
a2lx1 -| a22x2: b2.
If we interPret x1, x2 as coordinates in the xlxz-plane, then each of the two equations represents a straight line,
and (r1, x2) is a solution if and only if the point P with coordinates x1, x2lies on both lines. Hence there are
three possible cases:
(a/ Precisely one solution if the lines intersect.
(b) Infinitely many solutions if the lines coincide.
(c) No solution if the lines are parallel
For instance,
ExAMPLE I
lnf in iteIy
many solutions
No solution
Fig. t56. Three
equations in
three unknowns
interpreted as
planes in space
X1 1-X2=t
2xr-xr=O
Case (o)
xr+ xr- l
2x1+2xr=2
Case (ó)
x7+x,2=I
X1 1-X2=O
Case (c)
If the sYstem is homogenous, Case (c) cannot happen, because then those two straight lines pass through the
origin, whose coordinates 0, 0 constitute the trivial solution. If you wish, consider three equations in three
unknowns aS rePresentations of three planes in space and discuss the various possible cases in a similar fashion.
See Fig. 156.
Our simple example illustrates thatasystem (1) may perhaps have no solution. This poses
the following problem. Does a given system (1) have a solution? Under what conditions
does it have precisely one solution? If it has more than one solution, how can we
characterize the Set of all solutions? How can we actually obtain the solutions? perhaps
the last question is the most immediate one from a píactical viewpoint. We shall answer
it first and discuss the other questions in Sec. 7.5.
Gauss Elimination and Back substitution
This is a standard elimination method for solving linear systems that proceeds
SYstematically irrespective of particular features of the coefficients. It is a method of great
Practical importance and is reasonable with respect to computing time and storage demand
(two aspects we shall consider in Sec. 20.1 in the chapter on numeric linear algebra). We
begin by motivating the method. If a system is in ..triangular form,o' say,
2x1 l 5rr: 2
I3xr: _26
We can solve it by "back substitutionr" that is, solve the last equation for the variable,
Xz: -26/13 : -2, and then work backward, substitutin1 xz - -2 into the first equation
l
Unique soIution