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.
| 
