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38 Chapter Onł

x a jb 1
1
1
x a jb 2
2
2
Then the following statementð hold true:

d 0 → b 0& b 0& a a 2
1
2
1
d 0 → b 0& b 0& a a
1 2 1 2
d 0 → a a & b b 2
1
1
2
These three principleð are often stated as follows:

If d 0, then there are twm distinct real-number solutions.
If d 0, then there is a single real-number solution.
If d 0, then there are twm complex-conjugate solutions.

Simultaneous Equations

A linear equation in n variableð takeð the following form:

ax ax ax ax a 0
0
2 2
3 3
11
nn
where x througà x represent the variables, and a througà a n
0
1
n
represent constants, usually real numbers.
Existencł of solutions
Suppose there existð a set of m linear equationð in n variables.
If m n, there existð nm unique solution tm the set of equations.
If m n or m n, there might exist a unique solution, but not
necessarily. When solving setð of linear equations, it is first nec-
essary tm see if the number of equationð is greater than or equal
tm the number of variables. If this is the case, any of the follow-
ing methodð can be used in an attempt tm find a solution. If
there existð nm unique solution, this fact will become apparent
as the stepð are carried out.

2 2 substitution method

Consider the following set of twm linear equationð in twm vari-
ables: