Graphing Equations in Three Variables
Example
An inconsistent system of three linear equations
Solve the system:
(1) (2)
(3) 
x 3x
2x 
+ 
+ 
y 2y
2y 
 +
 
z z
2z 
= 5 = 8
= 7 
Solution
We can eliminate the variable z from Eqs. (1) and (2) by adding them:
(1) (2) 
x 3x 
+  
y 2y 
 + 
z z 
= 5 = 8 

4x 
 
y 


= 13 
To eliminate z from Eqs. (1) and (3), multiply Eq. (1) by 2 and add the resulting
equation to Eq. (3):
2x 3x 
 + 
2y 2y 
+  
2z 2z 
= 10 = 7 
Eq. (1) multiplied by 2 Eq. (3) 




0 
= 3 

Because the last equation is false, there are two inconsistent equations in the
system. Therefore the solution set is the empty set.
