~~REVEAL~~
==== Linear equations in 3 variables ====

==== Definition ====

{{page>linear equation in 3 variables}}

==== Examples ====

  * $x+y+z=1$ 
  * {{::xyz1.png?nolink&600|}}

==== ====
  * $x+y=1$ This may be viewed as a linear equation in 3 variables, since it is equivalent to $x+y+0z=1$. 
  * {{::xy1.png?nolink&600|}}

==== ====
  * $z=1$, viewed as the equation $0x+0y+z=1$
  * {{::z1.png?nolink&600|}}

==== Linear equations (in general) ====

{{page>linear equation}}

==== Example ====

\[ 3x_1+5x_2-7x_3+11x_4=12\] is a linear equation in 4 variables. 

  * A typical solution will be a point $(x_1,x_2,x_3,x_4)\in \mathbb{R}^4$ so that $3x_1+5x_2-7x_3+11x_4$ really does equal $12$. 
  * For example, $(-2,0,-1,1)$ is a solution. 
  * The set of all solutions is a 3-dimensional object in $\mathbb{R}^4$, called a [[wp>hyperplane]]. 
  * Since we can't draw pictures in 4-dimensional space $\mathbb{R^4}$ we can't draw this set of solutions!

==== Systems of linear equations ====

{{page>system of linear equations}}

==== Example ====
Find the line of intersection of the two planes 
$ x+3y+z=5$ and $ 2x+7y+4z=17$.

  * {{::intersection.png?nolink&600|}}

==== Intersection of $ x+3y+z=5$ and $ 2x+7y+4z=17$ ====

  * To find the equation of the line of intersection, we must find the points which are solutions of //both// equations at the same time. 
  * Eliminating variables, we get $x=-16+5z$, $y=7-2z$
  * The line of intersection consists of the points $(-16+5z,7-2z,z)$, where $z\in\mathbb{R}$