(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 31843, 1030] NotebookOptionsPosition[ 26505, 846] NotebookOutlinePosition[ 27516, 881] CellTagsIndexPosition[ 27361, 874] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Linear Equations in Two Variables, Version D", "Title"], Cell["\<\ Kenneth M. Levasseur Department of Mathematical Sciences University of Massachusetts Lowell Lowell, MA 01854 Kenneth_Levasseur@uml.edu\ \>", "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Loading the LinearEquations Package ", "Section"], Cell[TextData[{ "You won't find a definition of ", StyleBox["EquationOfLine", "Input"], " in this notebook. At this stage it's been put into a package simply \ called ", ButtonBox["LinearEquations.nb", BaseStyle->"Hyperlink", ButtonData:>{"::AddOns:Applications:LinearEquations.nb", None}], ". You should open this notebook in order to look at it after reading this \ notebook. The hyperlink in this cell should work if you've placed the \ package into the AddOns/Applications directory of your ", StyleBox["Mathematica", FontSlant->"Italic"], " directory.\n\nThis is the best way to load a package. Needs will check \ the current value of Contexts[] and if the arguement isn't already listed, \ the appropriete package is loaded. " }], "Text"], Cell[BoxData[ RowBox[{"Needs", "[", "\"\\"", "]"}]], "Input"], Cell["\<\ If you're unfamiliar with a package you've just loaded, one way to start \ learning about it is to look at the names of the functions that are defined \ within it. Naturally, you would expect extensive documentation for large \ packages, and that is normally what you will find. This small package has no \ documentation.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "LinearEquations`*"}]], "Input"], Cell[BoxData[ DynamicModuleBox[{Typeset`open$$ = True}, PaneSelectorBox[{False-> RowBox[{ OpenerBox[Dynamic[Typeset`open$$], ImageSize->Small], StyleBox["LinearEquations`", "InfoHeading"]}], True->GridBox[{ { RowBox[{ OpenerBox[Dynamic[Typeset`open$$], ImageSize->Small], StyleBox["LinearEquations`", "InfoHeading"]}]}, {GridBox[{ { ButtonBox["EquationOfLine", BaseStyle->"InformationLink", ButtonData:>{ "Info3393649410-9503586", "LinearEquations`EquationOfLine"}, ButtonNote->"LinearEquations`"], ButtonBox["StandardFormOfLine", BaseStyle->"InformationLink", ButtonData:>{ "Info3393649410-9503586", "LinearEquations`StandardFormOfLine"}, ButtonNote->"LinearEquations`"], ButtonBox["XIntercept", BaseStyle->"InformationLink", ButtonData:>{ "Info3393649410-9503586", "LinearEquations`XIntercept"}, ButtonNote->"LinearEquations`"], ButtonBox["YIntercept", BaseStyle->"InformationLink", ButtonData:>{ "Info3393649410-9503586", "LinearEquations`YIntercept"}, ButtonNote->"LinearEquations`"]}, { ButtonBox["PlotLine", BaseStyle->"InformationLink", ButtonData:>{"Info3393649410-9503586", "LinearEquations`PlotLine"}, ButtonNote->"LinearEquations`"], ButtonBox["x", BaseStyle->"InformationLink", ButtonData:>{"Info3393649410-9503586", "LinearEquations`x"}, ButtonNote->"LinearEquations`"], ButtonBox["y", BaseStyle->"InformationLink", ButtonData:>{"Info3393649410-9503586", "LinearEquations`y"}, ButtonNote->"LinearEquations`"], ""} }, DefaultBaseStyle->"InfoGrid", GridBoxItemSize->{"Columns" -> {{ Scaled[0.2375]}}}]} }, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}]}, Dynamic[Typeset`open$$], ImageSize->Automatic]]], "Print", "InfoCell", CellChangeTimes->{3.39366381267745*^9}] }, Open ]], Cell["\<\ You can see from the list of names that in addition to EquationOfLine, there \ are a few other functions that one would expect to be related to linear \ equations. It's easy to see what an individual function does. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "XIntercept"}]], "Input"], Cell[BoxData[ StyleBox["\<\"XIntercept[line] is the x-intercept of line. Takes the form \ of a point {a,0}\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.393663814529488*^9}, CellTags->"Info3393649413-5678802"] }, Open ]], Cell["\<\ This is where the usage statments are crucial. They provide this \ documentation. Let's try this function\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"XIntercept", "[", RowBox[{"y", "==", RowBox[{ RowBox[{"5", "x"}], "-", "3"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ FractionBox["3", "5"], ",", "0"}], "}"}], TraditionalForm]], "Output", CellChangeTimes->{3.393663815084703*^9}] }, Open ]], Cell["\<\ Documentation isn't adequate if users get confused. If you thought the \ following would work, it's my fault. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"XIntercept", "[", RowBox[{ RowBox[{"5", "x"}], "-", "3"}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Solve", "::", "\<\"eqf\"\>"}], RowBox[{ ":", " "}], "\<\"\\!\\(TraditionalForm\\`\\(\\(\\(5\\\\ x\\)\\) - 3\\)\\) \ is not a well-formed equation. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/message/Solve/eqf\\\", ButtonNote -> \ \\\"Solve::eqf\\\"]\\)\"\>"}], TraditionalForm]], "Message", CellChangeTimes->{3.3936638154529743`*^9}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"ReplaceAll", "::", "\<\"reps\"\>"}], RowBox[{ ":", " "}], "\<\"\\!\\(TraditionalForm\\`\\({\\(\\(5\\\\ x\\)\\) - \ 3}\\)\\) is neither a list of replacement rules nor a valid dispatch table, \ and so cannot be used for replacing. \ \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/ReplaceAll/reps\\\", \ ButtonNote -> \\\"ReplaceAll::reps\\\"]\\)\"\>"}], TraditionalForm]], "Message", CellChangeTimes->{3.393663815863792*^9}], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"x", "/.", "\[InvisibleSpace]", RowBox[{ RowBox[{"5", " ", "x"}], "-", "3"}]}], ",", "0"}], "}"}], TraditionalForm]], "Output", CellChangeTimes->{3.393663815949301*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" Tests", "Section"], Cell["\<\ At each stage of development, you should continue to test your functions.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquationOfLine", "[", RowBox[{"5", ",", RowBox[{"-", "1"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"y", "\[LongEqual]", RowBox[{ RowBox[{"5", " ", "x"}], "-", "1"}]}], TraditionalForm]], "Output", CellChangeTimes->{3.393663816311557*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquationOfLine", "[", RowBox[{"5", ",", RowBox[{"-", "1"}], ",", RowBox[{"PlotLine", "->", "True"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[{{-1.8, -10.}, {-1.4133758129597247`, -8.066879064798623}, \ {-1.0212374100771209`, -6.106187050385604}, {-0.5960779053832947, \ -3.9803895269164737`}, {-0.1991217886741647, -1.9956089433708235`}, { 0.19003910331557547`, -0.049804483422122625`}, {0.612221097116538, 2.06110548558269}, {1.0061997029328045`, 4.030998514664022}, { 1.4331994105602932`, 6.165997052801466}, {1.8524038934683922`, 8.262019467341961}, {2.2, 10.}}]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesOrigin->{0, 0}, PlotRange->{{-10, 10}, {-10, 10}}, PlotRangeClipping->True, PlotRangePadding->{Automatic, Automatic}], TraditionalForm]], "Print", CellChangeTimes->{3.393663816625745*^9}], Cell[BoxData[ FormBox[ RowBox[{"y", "\[LongEqual]", RowBox[{ RowBox[{"5", " ", "x"}], "-", "1"}]}], TraditionalForm]], "Output", CellChangeTimes->{3.3936638167094803`*^9}] }, Open ]], Cell["\<\ You should be able to provide the default and things should still work.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquationOfLine", "[", RowBox[{ RowBox[{"-", "5.7"}], ",", "104.8", ",", RowBox[{"PlotLine", "->", "False"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"y", "\[LongEqual]", RowBox[{"104.8`", "\[InvisibleSpace]", "-", RowBox[{"5.7`", " ", "x"}]}]}], TraditionalForm]], "Output", CellChangeTimes->{3.3936638168262873`*^9}] }, Open ]], Cell["Undefined option values should revert to the default", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq", "=", RowBox[{"EquationOfLine", "[", RowBox[{ RowBox[{"-", "\[Pi]"}], ",", "4", ",", RowBox[{"PlotLine", "->", "Maybe"}]}], "]"}]}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"y", "\[LongEqual]", RowBox[{"4", "-", RowBox[{"\[Pi]", " ", "x"}]}]}], TraditionalForm]], "Output", CellChangeTimes->{3.393663816946498*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"XIntercept", "[", "eq", "]"}], ",", RowBox[{"YIntercept", "[", "eq", "]"}]}], "}"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { FractionBox["4", "\[Pi]"], "0"}, {"0", "4"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], TraditionalForm]], "Output", CellChangeTimes->{3.3936638176166677`*^9}] }, Open ]], Cell[TextData[{ "There was a problem with some equations with real constants not simplifying \ very well. ", StyleBox["StandardFormOfLine", "Input"], " will take care of that." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "StandardFormOfLine"}]], "Input"], Cell[BoxData[ StyleBox["\<\"StandardFormOfLine[line] returns the standard form of \ line.\"\>", "MSG"]], "Print", "PrintUsage", CellChangeTimes->{3.39366381910576*^9}, CellTags->"Info3393649418-4562407"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"StandardFormOfLine", "[", "eq", "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[ RowBox[{"\[Pi]", " ", "x"}], "4"], "+", FractionBox["y", "4"]}], "\[LongEqual]", "1"}], TraditionalForm]], "Output", CellChangeTimes->{3.393663819606394*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"StandardFormOfLine", "[", RowBox[{ RowBox[{"y", "+", "2.457"}], "==", RowBox[{ RowBox[{"3.45", " ", "x"}], " ", "+", "3.5"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"0.9587727708533076`", " ", "y"}], "-", RowBox[{"3.307766059443912`", " ", "x"}]}], "\[LongEqual]", "1"}], TraditionalForm]], "Output", CellChangeTimes->{3.393663819729464*^9}] }, Open ]], Cell["\<\ There had been a problem with not being able to reset the default of \ PlotLine. Now it works. The solution is described in LinearEquations.nb\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SetOptions", "[", RowBox[{"EquationOfLine", ",", RowBox[{"PlotLine", "->", "True"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"PlotLine", "\[Rule]", "True"}], "}"}], TraditionalForm]], "Output",\ CellChangeTimes->{3.393663819838737*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquationOfLine", "[", RowBox[{ RowBox[{"{", RowBox[{"3", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", RowBox[{"-", "1"}]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[{{0.5000000000000002, 10.}, {0.612221097116538, 9.775557805766924}, {1.0061997029328045`, 8.987600594134392}, { 1.4331994105602932`, 8.133601178879413}, {1.8524038934683922`, 7.295192213063215}, {2.243404988391795, 6.51319002321641}, { 2.6674271851264204`, 5.665145629747159}, {3.0632459938763494`, 4.873508012247301}, {3.451269577906889, 4.097460844186222}, { 3.8723142637486507`, 3.2553714725026985`}, {4.265155561605716, 2.4696888767885685`}, {4.691017961274004, 1.617964077451992}, { 5.088676972957597, 0.8226460540848066}, {5.478540759921798, 0.04291848015640376}, {5.901425648697223, -0.8028512973944455}, { 6.296107149487952, -1.5922142989759038`}, { 6.7238097520899025`, -2.447619504179805}, { 7.143717129972463, -3.2874342599449253`}, { 7.535421119870328, -4.0708422397406565`}, { 7.960146211579415, -4.9202924231588305`}, { 8.356667915303806, -5.713335830607612}, { 8.745394394308807, -6.490788788617614}, { 9.16714197512503, -7.334283950250061}, { 9.560686167956558, -8.121372335913115}, { 9.987251462599307, -8.974502925198614}, { 9.987450652118643, -8.974901304237285}, { 9.987649841637976, -8.975299683275953}, { 9.988048220676646, -8.976096441353292}, { 9.988844978753985, -8.97768995750797}, { 9.990438494908664, -8.980876989817329}, { 9.993625527218022, -8.987251054436044}, { 9.993824716737358, -8.987649433474715}, { 9.994023906256691, -8.988047812513383}, { 9.99442228529536, -8.988844570590722}, { 9.9952190433727, -8.9904380867454}, { 9.996812559527378, -8.993625119054755}, { 9.997011749046713, -8.994023498093426}, { 9.997210938566047, -8.994421877132094}, { 9.997609317604716, -8.995218635209433}, { 9.998406075682055, -8.99681215136411}, { 9.998605265201391, -8.997210530402782}, { 9.998804454720725, -8.99760890944145}, { 9.999202833759394, -8.998405667518789}, { 9.99940202327873, -8.99880404655746}, { 9.999601212798066, -8.999202425596131}, { 9.999800402317401, -8.999600804634802}, { 9.999999591836735, -8.99999918367347}}]}}, AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948], Axes->True, AxesOrigin->{0, 0}, PlotRange->{{-10, 10}, {-10, 10}}, PlotRangeClipping->True, PlotRangePadding->{Automatic, Automatic}], TraditionalForm]], "Print", CellChangeTimes->{3.393663819996282*^9}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"2", " ", "x"}], "+", "y"}], "\[LongEqual]", "11"}], TraditionalForm]], "Output", CellChangeTimes->{3.393663820099697*^9}] }, Open ]], Cell["I can actively suppress the plot now.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"EquationOfLine", "[", RowBox[{ RowBox[{"{", RowBox[{"3", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", RowBox[{"-", "1"}]}], "}"}], ",", RowBox[{"PlotLine", "->", "False"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"2", " ", "x"}], "+", "y"}], "\[LongEqual]", "11"}], TraditionalForm]], "Output", CellChangeTimes->{3.3936638202332563`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"SetOptions", "[", RowBox[{"EquationOfLine", ",", RowBox[{"PlotLine", "->", "False"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"PlotLine", "\[Rule]", "False"}], "}"}], TraditionalForm]], "Output", CellChangeTimes->{3.3936638204237413`*^9}] }, Open ]], Cell["Horizontal Line Tests", "Text", CellTags->"Horizontal"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"heq", "=", RowBox[{"EquationOfLine", "[", RowBox[{ RowBox[{"{", RowBox[{"2", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", "5"}], "}"}]}], "]"}]}], "\n"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"y", "\[LongEqual]", "5"}], TraditionalForm]], "Output", CellChangeTimes->{3.393663821073572*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"XIntercept", "[", "heq", "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"XIntercept", "::", "\<\"HorizontalLine\"\>"}], RowBox[{":", " "}], "\<\"Line is horizontal - 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