Graphing Equations in Three Variables
Example
A dependent system of three equations
Solve the system:
(1) (2)
(3) 
2x
6x
4x 

+
 
3y
9y
6y 

+
 
z 3z
2z 
= 4
= 12
= 8 
Solution
We will first eliminate x from Eqs. (1) and (2). Multiply Eq. (1) by 3 and add the
resulting equation to Eq. (2):
6x 6x 
 + 
9y 9y 
 + 
3z 3z 
= 12 = 12 
Eq. (1) multiplied by 3 Eq. (2) 




0 
= 0 

The last statement is an identity. The identity occurred because Eq. (2) is a multiple
of Eq. (1). In fact, Eq. (3) is also a multiple of Eq. (1). These equations are dependent.
They are all equations for the same plane. The solution set is the set of all
points on that plane,
{(x, y, z)  2x  3y  z = 4}.
Helpful Hint
If you recognize that multiplying
Eq. (1) by 3 will produce
Eq. (2), and multiplying Eq. (1)
by 2 will produce Eq. (3), then
you can conclude that all
three equations are equivalent
and there is no need to
add the equations.
