MST10030 notes wiki

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Linear equations in 3 variables

Definition

If $a,b,c,d$ are any fixed numbers, then equation \[ ax+by+cz=d\] is a linear equation in 3 variables.

When you draw the set of all solutions of a linear equation in 3 variables, you always get a plane in 3-dimensional space, $\mathbb{R}^3$.

Examples

$x+y+z=1$ <html><iframe scrolling=“no” src=“https://tube.geogebra.org/material/iframe/id/528999/width/800/height/503/border/888888/rc/true/ai/false/sdz/true/smb/false/stb/true/stbh/true/ld/false/sri/true/at/auto” width=“800px” height=“503px” style=“border:0px;”> </iframe></html>
$x+y=1$ This may be viewed as a linear equation in 3 variables, since it is equivalent to $x+y+0z=1$. <html><iframe scrolling=“no” src=“https://tube.geogebra.org/material/iframe/id/529043/width/800/height/503/border/888888/rc/true/ai/false/sdz/true/smb/false/stb/true/stbh/true/ld/false/sri/true/at/auto” width=“800px” height=“503px” style=“border:0px;”> </iframe></html>
$z=1$, viewed as the equation $0x+0y+z=1$ <html><iframe scrolling=“no” src=“https://tube.geogebra.org/material/iframe/id/529069/width/800/height/503/border/888888/rc/true/ai/false/sdz/true/smb/false/stb/true/stbh/true/ld/false/sri/true/at/auto” width=“800px” height=“503px” style=“border:0px;”> </iframe><br /></html>This plane is horizontal (parallel to the $x$-$y$ plane).