6.2 Linear Systems of Equations in Two
Variables
Solving systems by elimination
Example:
1
x+y=4
2
x- y=1
Strategy -- add the two equations in order to "eliminate" one of
the variables
+
1
x+y=4
2
x- y=1
3
2x
=5
You now have one equation with one unknown.
Solve it and finish as for substitution.
Example:
1
3x + 3y = 15
2 2x + 6y = 22
Mere adding won't eliminate. We multiply each equation by an
appropriate constant so adding will eliminate (either x or y):
1 x 2:
6x + 6y = 30
2 x -3: -6x - 18y = -66
-12y = -36
y=3
etc.
If all proceeds smoothly, and you get exactly one solution,
implying that the lines cross in exactly one point. The
system is termed consistent and independent.
6.2-1
Funny things that can happen, Ha! Ha! . . .
. . . and how to cope
Funny thing #1:
How to cope:
Example:
you get an equation that is never true, e.g.
10 = 0
STOP! and write the answer "no solution"
 system consists of two parallel lines (they
never cross)
 this is called an inconsistent system
x - y = 10
-x + y = 5
_________________________
0 + 0 = 15
no solution  write this
Funny thing #2:
How to cope:
Example:
you get an equation that is always true, e.g.
10 = 10
STOP! and write answer as shown in the
example below
 system consists of two identical lines
(every point of the first is a point on the
second)
 all (infinitely many) points on either line
are solutions
 this is called a dependent (but
consistent) system
x - y = 10
-x + y = -10
_________________
0+0= 0
{(x, y) | x - y = 10}  write this
6.2-2