NOTE When a system has an infinite number of solutions, as in Example 3(b),
either equation of the system could be used to write the solution set. We prefer to use
the equation in standard form with integer coefficients that have greatest common
factor 1.If neither of the given equations is in this form, we will use an equivalent
equation that is in standard form with integer coefficients that have greatest common
factor 1.
SECTION 4.1 Solving Systems of Linear Equations by Graphing 251
The system in Example 2has exactly one solution. A system with at least one
solution is called a consistent system.A system with no solution, such as the one in
Example 3(a),is called an inconsistent system.
The equations in Example 2are independent equationswith different graphs.
The equations of the system in Example 3(b)have the same graph and are equiva-
lent. Because they are different forms of the same equation, these equations are
called dependent equations.
Examples 2 and 3show the three cases that may occur when solving a system
of equations with two variables.
Three Cases for Solutions of Systems
1. The graphs intersect at exactly one point, which gives the (single) ordered-
pair solution of the system. The system is consistentand the equations are
independent.See FIGURE 5(a).
2. The graphs are parallel lines, so there is no solution and the solution set
is The system is inconsistentand the equations are independent.See
FIGURE 5(b).
3. The graphs are the same line. There is an infinite number of solutions, and the
solution set is written in set-builder notation as
where one of the equations is written after the symbol. The system is
consistentand the equations are dependent.See FIGURE 5(c).
|
51 x, y 2 | 6 ,
0.
x
y
0
One
solution
x
y
0
No
solution
x
y
0
Infinite
number of
solutions
(a) (b) (c)
FIGURE 5
OBJECTIVE 4 Identify special systems without graphing.Example 3
showed that the graphs of an inconsistent system are parallel lines and the graphs of
a system of dependent equations are the same line. We can recognize these special
kinds of systems without graphing by using slopes.
