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--- lecture_2_sides [2016/01/24 16:24] – rupert
+++ lecture_2_sides [2016/01/24 16:44] (current) – rupert
@@ Line 2: / Line 2: @@
 ==== Linear equations in 3 variables ====
-=== Definition ===
+==== Definition ====
 {{page>linear equation in 3 variables}}
-=== Examples ===
+==== 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 }}
   * $x+y+z=1$ <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).
@@ Line 19: / Line 20: @@
 {{page>linear equation}}
-=== Example ===
+==== Example ====
 \[ 3x_1+5x_2-7x_3+11x_4=12\] is a linear equation in 4 variables.
@@ Line 32: / Line 33: @@
 {{page>system of linear equations}}
-=== Example ===
+==== Example ====
 Find the line of intersection of the two planes
-\[ x+3y+z=5\] and \[ 2x+7y+4z=17.\]
+$ 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:
+  * <html><iframe scrolling="no" src="https://tube.geogebra.org/material/iframe/id/529147/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>
-<html><iframe scrolling="no" src="https://tube.geogebra.org/material/iframe/id/529147/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>
+==== 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
+  * To find the equation of the line of intersection, we must find the points which are solutions of //both// equations at the same time.
-\[ x=-16+5z,\quad y=7-2z\]
+  * Eliminating variables, we get $x=-16+5z$, $y=7-2z$
-which tells us that for any value of $z$, the point
+  * The line of intersection consists of the points $(-16+5z,7-2z,z)$, where $z\in\mathbb{R}$
-\[ (-16+5z,7-2z,z)\]
-is a typical point in the line of intersection.