==== Linear equations in 3 variables ==== === Definition === {{page>linear equation in 3 variables}} === Examples === Note: you can view the examples below from different angles, by clicking the "Rotate 3D graphics view" button. {{ screenshot_from_2015-01-22_10_40_28.png?nolink }} * $x+y+z=1$ * $x+y=1$ This may be viewed as a linear equation in 3 variables, since it is equivalent to $x+y+0z=1$. * $z=1$, viewed as the equation $0x+0y+z=1$
This plane is horizontal (parallel to the $x$-$y$ plane). ==== 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 \[ 2x+7y+4z=17.\] Just to get an idea of what's going on, here's a picture of the two planes: 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 \[ x=-16+5z,\quad y=7-2z\] which tells us that for any value of $z$, the point \[ (-16+5z,7-2z,z)\] is a typical point in the line of intersection.