4.4
Systems of Equations - Three Variables
Objective: Solve systems of equations with three variables
using addi-
tion/elimination.
Solving systems of equations with 3 variables is very simila
r to how we solve sys-
tems with two variables. When we had two variables we reduced
the system down
to one with only one variable (by substitution or addition).
With three variables
we will reduce the system down to one with two variables (usua
lly by addition),
which we can then solve by either addition or substitution.
To reduce from three variables down to two it is very importan
t to keep the work
organized. We will use addition with two equations to elimin
ate one variable.
This new equation we will call (A). Then we will use a different
pair of equations
and use addition to eliminate the
same
variable. This second new equation we
will call (B). Once we have done this we will have two equation
s (A) and (B)
with the same two variables that we can solve using either met
hod. This is shown
in the following examples.
Example 1.
3
x
+ 2
y
−
z
=
−
1
−
2
x
−
2
y
+ 3
z
= 5
We will eliminate
y
using two different pairs of equations
5
x
+ 2
y
−
z
=