Inconsistent Systems of Linear Equations
What does it mean when we say that a system of
linear equations is inconsistent? In order to understand this,
we must first recall what a solution of a sytem
is.
Definition A solution
of a system of linear equations in
one or more unknowns is a set of values for these unknowns that satisfies
all the equations in the system.
Example consider the system
Choosing x = 1/3 and y
= -2/3 and substituing them into the equations, we see
that both equations indeed give identities, hence the pair of values (x,
y) = (1/3, -2/3) is a solution for the system.
Definition A system of linear equations
is called inconsistent when it has
no solutions.
Example Consider the system
Clearly, there are no values for x
and y that satisfy both equations;
2x + y cannot be both
0 and 1.
What does this mean geometrically?
In this example, each of the equations describes a straight line - and
the lines are parallel! That is, there are no points at which the lines
intercept each other, and therefore no solutions to the system containing
both equations.