## Systems of Linear Equations in Three Variables

 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} 4x-2y+z&=2\quad \\\\ 6x+7y-z&=3 \\\\ 5x+y+2z&=7 \end{aligned} }

 $$\blacksquare$$ Geometrically, a linear equation in three variables forms a plane in a three-dimensional 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