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| lecture_2 [2015/01/22 10:39] – created rupert | lecture_2 [2015/01/22 11:07] (current) – [Linear equations (in general)] rupert |
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| === Examples === | === Examples === |
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| | 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 }} |
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| * $x+y+z=1$ <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> | * $x+y+z=1$ <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> | * $x+y=1$ 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). | * $z=1$, 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). |
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| | ==== Linear equations (in general) ==== |
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| | {{page>linear equation}} |
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| | === Example === |
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| | \[ 3x_1+5x_2-7x_3+11x_4=12\] is a linear equation in 4 variables. |
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| | * 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! |
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| | ==== Systems of linear equations ==== |
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| | {{page>system of linear equations}} |
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| | === Example === |
| | Find the line of intersection of the two planes |
| | \[ x+3y+z=5\] and \[ 2x+7y+4z=17.\] |
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| | Just to get an idea of what's going on, here's a picture of the two planes: |
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| | <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> |
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| | 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. |
