3.1 
Systems of Linear Equations in Three Variables


\(\blacksquare\) Two or more linear equations involving the same set of variables form a system of linear equations. 

\(\blacksquare\) The characteristics of systems of linear equations in three variables:
 Has three variables in each linear equation.
 The highest power of each variable is \(1\).


Example of system of linear equations in three variables: 

\(\boxed{\begin{aligned} 4x2y+z&=2\quad \\\\ 6x+7yz&=3 \\\\ 5x+y+2z&=7 \end{aligned} }\)

\(\blacksquare\) Geometrically, a linear equation in three variables forms a plane in a threedimensional space. 


\(\blacksquare\) There are three types of solutions for the systems of linear equations in three variables: 

One Solution 
Infinite
solutions 
No solution 
The planes
intersect at only
one point 
The planes
intersect in a
straight line 
The planes do
not intersect at
any point 

Methods used to solve systems of linear equations in three variables:
 Elimination method
 Substitution method


