Geometric Interpretation of Linear Equations: Intersection of Two Lines

2014-11-09 16:18

This GeoGebra applet demonstrates the geometric properties of linear equations. By defining the slopes and y-intercepts of two lines, it plots them on a Cartesian coordinate system. Using the Intersect command, it visually determines the intersection point of the two lines and dynamically displays its x and y coordinates. This example vividly illustrates that the solution to a system of two linear

#coordinate-geometry #function #graphing

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.18.0" id="b0987826-c29c-40eb-8b1b-2b6b85445168"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1616" height="876" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.12376237623762376" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="450,155,700,550" />
		<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 22 , 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
		<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
		<view id="8" toolbar="1001 | 1002 | 1003  || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
		<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1384" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
		<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
	<input show="true" cmd="true" top="false" />
	<dockBar show="true" east="true" />
</perspective>
	</perspectives>
	<labelingStyle  val="0"/>
	<font  size="12"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<size  width="1384" height="682"/>
	<coordSystem xZero="862.0" yZero="503.99999999999994" scale="49.99999999999997" yscale="50.00000000000002"/>
	<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
	<bgColor r="255" g="255" b="255"/>
	<axesColor r="0" g="0" b="0"/>
	<gridColor r="192" g="192" b="192"/>
	<lineStyle axes="1" grid="10"/>
	<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0"/>
	<coordStyle val="0"/>
	<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="m_1">
	<value val="1.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="137.0" fixed="true" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b_1">
	<value val="0.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="183.0" fixed="true" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="line_1" exp="y = (m_1 * x) + b_1" type="line" />
<element type="line" label="line_1">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="-1.0" y="1.0" z="0.0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eqnStyle style="explicit"/>
</element>
<element type="numeric" label="m_2">
	<value val="-1.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="289.0" fixed="false" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b_2">
	<value val="2.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="336.0" fixed="false" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="link_2" exp="y = (m_2 * x) + b_2" type="line" />
<element type="line" label="link_2">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="1.0" y="1.0" z="-2.0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eqnStyle style="explicit"/>
</element>
<expression label="文本1" exp="&quot;Line 1: &quot; + line_1 + &quot;&quot;" />
<element type="text" label="文本1">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="false" sizeM="1.4" size="5" style="1"/>
	<startPoint  x="-13.920000000000007" y="8.259999999999994" z="1.0"/>
</element>
<expression label="文本2" exp="&quot;Line2: &quot; + link_2 + &quot;&quot;" />
<element type="text" label="文本2">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="false" sizeM="1.4" size="5" style="1"/>
	<startPoint  x="-13.920000000000009" y="5.239999999999996" z="1.0"/>
</element>
<command name="Intersect">
	<input a0="line_1" a1="link_2"/>
	<output a0="A"/>
</command>
<element type="point" label="A">
	<show object="true" label="true"/>
	<objColor r="68" g="68" b="68" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="2.0" y="2.0" z="2.0"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<expression label="xcoordinate" exp="x(A)" />
<element type="numeric" label="xcoordinate">
	<value val="1.0"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="ycoordinate" exp="y(A)" />
<element type="numeric" label="ycoordinate">
	<value val="1.0"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="文本3" exp="&quot;Solution: x = &quot; + xcoordinate + &quot;&#10;              　y = &quot; + ycoordinate + &quot;&quot;" />
<element type="text" label="文本3">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="false" sizeM="1.4" size="5" style="1"/>
	<startPoint  x="-13.94" y="1.6200000000000006" z="1.0"/>
</element>
</construction>
</geogebra>

Geometric Interpretation of Linear Equations: Intersection of Two Lines

Loading the GeoGebra board and drawing, please wait…

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.18.0" id="b0987826-c29c-40eb-8b1b-2b6b85445168"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1616" height="876" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.12376237623762376" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="450,155,700,550" />
		<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 551 550 11 , 22 , 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
		<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
		<view id="8" toolbar="1001 | 1002 | 1003  || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
		<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1384" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
		<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
		<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
	<input show="true" cmd="true" top="false" />
	<dockBar show="true" east="true" />
</perspective>
	</perspectives>
	<labelingStyle  val="0"/>
	<font  size="12"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<size  width="1384" height="682"/>
	<coordSystem xZero="862.0" yZero="503.99999999999994" scale="49.99999999999997" yscale="50.00000000000002"/>
	<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
	<bgColor r="255" g="255" b="255"/>
	<axesColor r="0" g="0" b="0"/>
	<gridColor r="192" g="192" b="192"/>
	<lineStyle axes="1" grid="10"/>
	<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0"/>
	<coordStyle val="0"/>
	<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="m_1">
	<value val="1.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="137.0" fixed="true" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b_1">
	<value val="0.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="183.0" fixed="true" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="line_1" exp="y = (m_1 * x) + b_1" type="line" />
<element type="line" label="line_1">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="-1.0" y="1.0" z="0.0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eqnStyle style="explicit"/>
</element>
<element type="numeric" label="m_2">
	<value val="-1.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="289.0" fixed="false" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="b_2">
	<value val="2.0"/>
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
	<layer val="0"/>
	<labelMode val="1"/>
	<slider min="-5" max="5" absoluteScreenLocation="true" width="200.0" x="215.0" y="336.0" fixed="false" horizontal="true"/>
	<lineStyle thickness="10" type="0" typeHidden="1"/>
	<animation step="1" speed="1" type="0" playing="false"/>
</element>
<expression label="link_2" exp="y = (m_2 * x) + b_2" type="line" />
<element type="line" label="link_2">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="1.0" y="1.0" z="-2.0"/>
	<lineStyle thickness="2" type="0" typeHidden="1"/>
	<eqnStyle style="explicit"/>
</element>
<expression label="文本1" exp="&quot;Line 1: &quot; + line_1 + &quot;&quot;" />
<element type="text" label="文本1">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="false" sizeM="1.4" size="5" style="1"/>
	<startPoint  x="-13.920000000000007" y="8.259999999999994" z="1.0"/>
</element>
<expression label="文本2" exp="&quot;Line2: &quot; + link_2 + &quot;&quot;" />
<element type="text" label="文本2">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="false" sizeM="1.4" size="5" style="1"/>
	<startPoint  x="-13.920000000000009" y="5.239999999999996" z="1.0"/>
</element>
<command name="Intersect">
	<input a0="line_1" a1="link_2"/>
	<output a0="A"/>
</command>
<element type="point" label="A">
	<show object="true" label="true"/>
	<objColor r="68" g="68" b="68" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<coords x="2.0" y="2.0" z="2.0"/>
	<pointSize val="3"/>
	<pointStyle val="0"/>
</element>
<expression label="xcoordinate" exp="x(A)" />
<element type="numeric" label="xcoordinate">
	<value val="1.0"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="ycoordinate" exp="y(A)" />
<element type="numeric" label="ycoordinate">
	<value val="1.0"/>
	<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="文本3" exp="&quot;Solution: x = &quot; + xcoordinate + &quot;&#10;              　y = &quot; + ycoordinate + &quot;&quot;" />
<element type="text" label="文本3">
	<show object="true" label="true"/>
	<objColor r="0" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<font serif="false" sizeM="1.4" size="5" style="1"/>
	<startPoint  x="-13.94" y="1.6200000000000006" z="1.0"/>
</element>
</construction>
</geogebra>