folder for website

LaTeX/.gitattributes

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+# Auto detect text files and perform LF normalization
+* text=auto

LaTeX/.ipynb_checkpoints/Untitled-checkpoint.ipynb

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+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

LaTeX/.ipynb_checkpoints/answers-checkpoint.tex

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+% A (minimal) template for problem sets and solutions using the exam document class
+
+% Organization:
+%% Define new commands, macros, etc. in macros.tex
+%% Anything that you would put before \begin{document} should go in prelude.tex
+
+%% For multiple psets, each should get its own file to \input into main with a \section{}
+
+\documentclass[answers]{exam}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{hyperref}
+\usepackage{multicol}
+
+
+\setlength\dottedlinefillheight{.5in}
+
+\title{The elimination method solving systems of linear equations}
+\author{\href{https://jordanbell.info/}{jordanbell.info}}
+\date{}
+\begin{document}
+
+\maketitle
+
+\thispagestyle{empty}
+
+\begin{questions}
+\question{Find the point of intersection of the two lines
+\begin{gather*}
+4x-7y=19\\
+3x-2y=11
+\end{gather*}
+(Nelson, {\em Foundations of Mathematics 10}, Section 5.3, page 217, question 3d)
+}
+
+\fillwithdottedlines{\stretch{1}}
+
+\newpage
+
+\begin{solution}
+Given
+\begin{gather*}
+4x-7y=19\\
+3x-2y=11
+\end{gather*}
+Multiply the first equation by 3
+\begin{gather*}
+12x-21y=57\\
+3x-2y=11
+\end{gather*}
+Muliply the second equation by $-4$
+\begin{gather*}
+12x-21y=57\\
+-12x+8y=-44
+\end{gather*}
+Add the two equations together
+\begin{gather*}
+12x-12x-21y+8y=57-44
+\end{gather*}
+Collect like terms
+\begin{gather*}
+-13y=13
+\end{gather*}
+Isolate $y$
+\begin{gather*}
+y=-1
+\end{gather*}
+Use $y=-1$ with the equation $3x-2y=11$
+\begin{gather*}
+3x-2(-1)=11
+\end{gather*}
+\begin{gather*}
+3x+2=11
+\end{gather*}
+Isolate $3x$
+\begin{gather*}
+3x=9
+\end{gather*}
+Isolate $x$
+\begin{gather*}
+x=3
+\end{gather*}
+The POI of the two lines is
+\begin{gather*}
+(3,-1)
+\end{gather*}
+\end{solution}
+
+\newpage
+
+\question{Find the point of intersection of the two lines
+\begin{gather*}
+5x+2y=48\\
+x+y=15
+\end{gather*}
+(Nelson, {\em Foundations of Mathematics 10}, Section 5.3, page 217, question 4c)
+}
+
+\fillwithdottedlines{\stretch{1}}
+
+\newpage
+
+\begin{solution}
+Given
+\begin{gather*}
+5x+2y=48\\
+x+y=15
+\end{gather*}
+Multiply the second equation by $-2$
+\begin{gather*}
+5x+2y=48\\
+-2x-2y=-30
+\end{gather*}
+Add the two equations together
+\begin{gather*}
+5x-2x+2y-2y=48-30
+\end{gather*}
+Collect like terms
+\begin{gather*}
+3x=18
+\end{gather*}
+Isolate $x$
+\begin{gather*}
+x=6
+\end{gather*}
+Use $y=6$ with the equation $x+y=15$
+\begin{gather*}
+6+y=15
+\end{gather*}
+Isolate $y$
+\begin{gather*}
+y=9
+\end{gather*}
+The POI of the two lines is
+\begin{gather*}
+(6,9)
+\end{gather*}
+\end{solution}
+
+\newpage
+
+\question{Find the point of intersection of the two lines
+\begin{gather*}
+x+2y=2\\
+2x+y=1
+\end{gather*}
+}
+
+\fillwithdottedlines{\stretch{1}}
+
+\newpage
+
+\begin{solution}
+Given
+\begin{gather*}
+x+2y=2\\
+2x+y=1
+\end{gather*}
+Multiply first equation by 2
+\begin{gather*}
+2x+4y=4\\
+2x+y=1
+\end{gather*}
+Multiply second equation by $-1$
+\begin{gather*}
+2x+4y=4\\
+-2x-y=-1
+\end{gather*}
+Add the two equations
+\begin{gather*}
+2x-2x+4y-y=4-1
+\end{gather*}
+Collect like terms
+\begin{gather*}
+3y=3
+\end{gather*}
+Isolate $y$
+\begin{gather*}
+y=1
+\end{gather*}
+Use $y=1$ with $2x+y=1$
+\begin{gather*}
+2x+1=1
+\end{gather*}
+Isolate $2x$
+\begin{gather*}
+2x=0
+\end{gather*}
+Isolate $x$
+\begin{gather*}
+x=0
+\end{gather*}
+The POI of the two lines is
+\begin{gather*}
+(0,1)
+\end{gather*}
+\end{solution}
+
+\newpage
+
+\question{Find the point of intersection of the two lines
+\begin{gather*}
+5x-4y=11\\
+-3x-13y=6
+\end{gather*}
+}
+
+\fillwithdottedlines{\stretch{1}}
+
+\newpage
+
+\begin{solution}
+Given
+\begin{gather*}
+5x-4y=11\\
+-3x-13y=6
+\end{gather*}
+Multiply first equation by 3:
+\begin{gather*}
+15x-12y=33\\
+-3x-13y=6
+\end{gather*}
+Multiply second equation by 5
+\begin{gather*}
+15x-12y=33\\
+-15x-65y=30
+\end{gather*}
+Add the two equations
+\begin{gather*}
+15x-15x-12y-65y=33+30
+\end{gather*}
+Collect like terms
+\begin{gather*}
+-77y=63
+\end{gather*}
+Isolate $y$
+\begin{gather*}
+y=\frac{63}{-77}
+\end{gather*}
+Write fractions in lowest terms (common factor $7$)
+\begin{gather*}
+y=-\frac{9}{11}
+\end{gather*}
+Use $y=-\frac{9}{11}$ with $5x-4y=11$
+\begin{gather*}
+5x-4\left(-\frac{9}{11}\right)=11
+\end{gather*}
+\begin{gather*}
+5x+\frac{36}{11}=11
+\end{gather*}
+Isolate $5x$
+\begin{gather*}
+5x=11-\frac{36}{11}
+\end{gather*}
+Make common denominator
+\begin{gather*}
+5x=\frac{121}{11}-\frac{36}{11}
+\end{gather*}
+Combine like terms
+\begin{gather*}
+5x=\frac{85}{11}
+\end{gather*}
+Isolate $x$
+\begin{gather*}
+x=\left(\frac{1}{5}\right) \left(\frac{85}{11}\right)
+\end{gather*}
+Fraction multiplication properties
+\begin{gather*}
+x=\left(\frac{85}{5}\right) \left(\frac{1}{11}\right)
+\end{gather*}
+\begin{gather*}
+x=17 \left(\frac{1}{11}\right)
+\end{gather*}
+\begin{gather*}
+x=\frac{17}{11}
+\end{gather*}
+The POI of the two lines is
+\begin{gather*}
+\left(\frac{17}{11}, -\frac{9}{11}\right)
+\end{gather*}
+\end{solution}
+\end{questions}
+
+\end{document}