Solving Systems of Linear
Equations in Three Variables
In Maths it is many times necessary to solve systems of linear equations, such as
x 
+ 
2y 
+ 
3z 
= 
1 
2x 
+ 
5y 
+ 
2z 
= 
2 
x 
+ 
y 
+ 
z 
= 
3 
There are are at least three ways to solve this set of equations: Elimination of variables,
Gaussian reduction, and Cramerâ€™s rule. The first approach is described below.
Elimination of Variables
In the example, you first eliminate x from the second two equations, by subtracting twice the first equation from the second, and subtracting the first equation from the third. The three equations then become
x 
+ 
2y 
+ 
3z 
= 
1 


y 
 
4z 
= 
0 


y 
 
2z 
= 
2 
Next, y is eliminated from the third equation, by adding the (new) second equation to the third, yielding
x 
+ 
2y 
+ 
3z 
= 
1 


y 
 
4z 
= 
0 



 
6z 
= 
2 
From the third equation, we conclude that
From the second equation, we conclude that
Finally, from the first equation, we find that
