5.2
Systems of Linear Equations in Three Unknowns
Objectives:
After completing this section, you should be able to:
1.
Solve linear systems in three unknowns using Gaussian elimination
2.
Use systems of linear equations in three unknowns to model and solve application problems
Keep This in Mind
A linear equation in three variables is an equation that can be put in the form
d
cz
by
ax
where
x
,
y
,
and
z
are the variables,
d
c
b
a
and
,
,
,
are real numbers; and
c
b
a
and
,
,
are not all zero at the same time.
The number
a
is the
leading coefficient
, and
x
is the
leading variable.
The graph of a linear equation in three variables is a plane in a three
dimensional coordinate system.
An example of what a plane would look like is
a floor or a desk top.
In general, we are interested in solving linear systems in three variables of this
type
3
3
3
3
2
2
2
2
1
1
1
1
d
z
c
y
b
x
a
d
z
c
y
b
x
a
d
z
c
y
b
x
a
where
3
2
1
3
2
1
3
2
1
3
2
1
and
,
,
,
,
,
,
,
,
,
,
,
d
d
d
c
c
c
b
b
b
a
a
a
are all real numbers.
By a
solution
to a system
of three linear equations in three variables we mean
an
ordered triple
that makes ALL three equations true. Geometrically, it is
where the graphs of the three equations intersect, what they have in common.
Just as with linear systems in two variables, there are three possible outcomes
that you may encounter when working with systems of linear equations in three
variables:
1.
exactly one or unique solution
2.
no solution
3.
infinitely many solutions
The terms
consistent
,
inconsistent
,
dependent
, and
independent
are used to
describe these solutions, just as they are for linear systems with two variables.