~~REVEAL~~ ==== Linear equations in 3 variables ==== ==== Definition ==== {{page>linear equation in 3 variables}} ==== Examples ==== * $x+y+z=1$ ==== ==== * $x+y=1$ 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$. * ==== 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 $z\in\mathbb{R}$