From 723b6ded0066651ac8521ce1320371c8430b1434 Mon Sep 17 00:00:00 2001
From: Christopher Baines <mail@cbaines.net>
Date: Tue, 1 Apr 2014 19:10:41 +0100
Subject: Initial commit

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+\documentclass[12pt,a4paper,twocolumn,landscape]{exam}
+
+\usepackage{xparse}
+\usepackage{tikz}
+    \usetikzlibrary{arrows,calc,fit}
+\usepackage{mathtools}
+\usepackage{fullpage}
+\usepackage{todonotes}
+\usepackage{float}
+\usepackage[compact,explicit]{titlesec} % http://ctan.org/pkg/titlesec
+\usepackage[utf8]{inputenc}
+\usepackage{titling}
+\usepackage{wrapfig}
+\usepackage{boxedminipage}
+\usepackage{parskip} % Don't indent paragraphs
+
+\usepackage{fib}
+\usepackage{common}
+
+\pagestyle{headandfoot}
+
+\firstpageheadrule
+\firstpageheader{Solving Linear Equations}
+                {Level 7-8}
+                {\today}
+
+\runningheadrule
+\runningheader{Solving Linear Equations}
+              {Solving Linear Equations, Page \thepage\ of \numpages}
+              {\today}
+
+\firstpagefooter{}{}{}
+\runningfooter{}{}{}
+
+\begin{document}
+
+\section*{Solving Linear Equations - Level 7-8}
+
+\begin{minipage}{\linewidth}
+
+\begin{wrapfigure}{R}{0.3\linewidth}
+    \centering
+    \begin{boxedminipage}{\linewidth}
+        Keywords:
+        \begin{itemize}
+            \item Linear
+            \item Solve
+            \item Unknown
+        \end{itemize}
+    \end{boxedminipage}
+\end{wrapfigure}
+
+Linear equations are written with one unknown variable which is shown by a
+letter.
+
+It is necessary to \textbf{rearrange} the equation in order to find the value
+of x.
+
+\end{minipage}
+
+\begin{questions}
+
+\question Solve 2x+3 = 7 for the unknown.
+
+\begin{minipage}{\linewidth}
+
+    \begin{wrapfigure}{R}{0.3\linewidth}
+        \vspace{-25pt}
+        \centering
+        \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+            thick,main node/.style={draw=none,fill=none}]
+
+            \node[main node] (1) {$2x + 3 = 7$};
+            \node[main node] (2) [below of=1] {$2x = 4$};
+
+            \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$-3$} (2.west)
+            (1.east) edge [bend left] node[right] {$-3$} (2.east);
+        \end{tikzpicture}
+    \end{wrapfigure}
+
+    Start by looking at the section of the equation that contains the unknown. In
+    this case the left hand side.
+
+\end{minipage}
+
+The next step is to collect all constants (numbers) on the side of the equation
+that does not contain the unknown. We want to move the 3 to the right hand
+side. Because it is +3, to move it to the other side we must -3.
+
+\begin{minipage}{\linewidth}
+
+    \begin{wrapfigure}{R}{0.3\linewidth}
+        \centering
+        \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+            thick,main node/.style={draw=none,fill=none}]
+
+            \node[main node] (1) {$2x = 4$};
+            \node[main node] (2) [below of=1] {$x = 2$};
+
+            \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$\div 2$} (2.west)
+            (1.east) edge [bend left] node[right] {$\div 2$} (2.east);
+        \end{tikzpicture}
+    \end{wrapfigure}
+
+    We can see that there are still numbers on the right hand side with the
+    unknown. The coefficient (multiplier) of x is 2. Since the 2 is times x we
+    need to divide by 2 to get the x by itself.
+
+\end{minipage}
+
+We can see that there are no more numbers on the side of x so the solution is:
+$x = 2$
+
+To check the solution, put the unknown back into the original equation. $2
+\times 2 + 3 = 7$
+
+\begin{keypoints}[t]
+    Key things to remember:
+    \begin{itemize}
+        \item Collect all \textbf{terms} involving the \textbf{unknown} to one
+              side of the equation.
+        \item The opposite of addition is subtraction.
+        \item The opposite of multiplication is division.
+        \item If you do something to one side of the equation you MUST do the
+              same thing to the other side.
+    \end{itemize}
+\end{keypoints}
+
+\subsection*{Higher Level Questions}
+
+A harder question is shown below, the steps taken are shown on the arrows.  The
+first step is to bring all \textbf{unknowns} to the same side of the equation.
+
+\question Find the value of x: $6x - 12 = x + 8$
+
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+        thick,main node/.style={draw=none,fill=none}]
+
+        \node[main node] (1) {$6x - 12 = x + 8$};
+        \node[main node] (2) [below of=1] {$5x - 12 = 8$};
+        \node[main node] (3) [below of=2] {$5x = 20$};
+        \node[main node] (4) [below of=3] {$x = 4$};
+
+        \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$-x$} (2.west)
+            (1.east) edge [bend left] node[right] {$-x$} (2.east)
+            (2.west) edge [bend right] node[left]{$+12$} (3.west)
+            (2.east) edge [bend left] node[right] {$+12$} (3.east)
+            (3.west) edge [bend right] node[left]{$\div 5$} (4.west)
+            (3.east) edge [bend left] node[right] {$\div 5$} (4.east);
+    \end{tikzpicture}
+\end{figure}
+
+Check:
+
+On the left hand side of the \textbf{original} equation:\\
+$6(4) - 12 = 24 - 12 = 12$
+
+On the right hand side of the \textbf{original} equation:\\
+$4 + 8 = 12$
+
+The left hand side and right hand side of the equation are equal so you know
+that your solution is correct: $x = 4$
+
+\end{questions}
+
+\newpage
+
+\subsection*{Practice Questions}
+
+\begin{questions}
+
+\question Find the value of x: $6+2x = x-6$
+
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+        thick,main node/.style={draw=none,fill=none}]
+
+        \node[main node] (1) {$6 + 2x = x - 6$};
+        \node[main node] (2) [below of=1] {$\fib{6 + x} = -6$};
+        \node[main node] (3) [below of=2] {$\fib{x} = \fib{-12}$};
+
+        \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$-x$} (2.west)
+            (1.east) edge [bend left] node[right] {$-x$} (2.east)
+            (2.west) edge [bend right] node[left]{$-6$} (3.west)
+            (2.east) edge [bend left] node[right] {$\fib{-6}$} (3.east);
+    \end{tikzpicture}
+\end{figure}
+
+Check:
+
+$6 + \fib{2}x = 6 + 2 \times -12 = \fib{-18}$\\
+$\fib{x} - 6 = \fib{-12} - 6 = \fib{-18}$
+
+\end{questions}
+
+\begin{questions}
+
+\newpage
+
+\subsection*{Solving Linear Equations}
+
+\question[3] Find the value of x: $6x + 13 = 4 + 9x$ \droppoints
+
+\begin{minipage}{0.69\linewidth}
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+        thick,main node/.style={draw=none,fill=none}]
+
+        \node[main node] (1) {$6x + 13 = 4 + 9x$};
+        \node[main node] (2) [below of=1] {$\fib{13} = 4 + 3x$};
+        \node[main node] (3) [below of=2] {$\fib{9} = \fib{3x}$};
+        \node[main node] (4) [below of=3] {$\fib{x} = \fib{3}$};
+
+        \path[every node/.style={font=\sffamily\small}]
+        (1.west) edge [bend right] node[left]{$-6x$} (2.west)
+        (1.east) edge [bend left] node[right] {$-6x$} (2.east)
+        (2.west) edge [bend right] node[left]{$-4$} (3.west)
+        (2.east) edge [bend left] node[right] {$-4$} (3.east)
+        (3.west) edge [bend right] node[left]{$\fib{\div 3}$} (4.west)
+        (3.east) edge [bend left] node[right] {$\fib{\div 3}$} (4.east);
+    \end{tikzpicture}
+\end{figure}
+\end{minipage}
+\begin{boxedminipage}{0.3\linewidth}
+    TIP:
+
+    When you have unknowns on both sides of the equations you can choose which
+    one to move.
+
+    Try and move the lower number of unknowns to avoid negatives.
+
+\end{boxedminipage}
+
+Check:
+
+$6\fib{x} + 13 = 6\fib{3} + 13 = 31$\\
+$4 + 9\fib{x} = 4 + \fib{27} = 31$
+
+\newpage
+
+\question[3] Find the value of x: $7x + 8 = 2x - 2$ \droppoints
+
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+        thick,main node/.style={draw=none,fill=none}]
+
+        \node[main node] (1) {$7x + 8 = 2x - 2$};
+        \node[main node] (2) [below of=1] {$5x + 8 = \fib{-2}$};
+        \node[main node] (3) [below of=2] {$\fib{5x} = \fib{-10}$};
+        \node[main node] (4) [below of=3] {$\fib{x} = \fib{-2}$};
+
+        \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$-2x$} (2.west)
+            (1.east) edge [bend left] node[right] {$-2x$} (2.east)
+            (2.west) edge [bend right] node[left]{$\fib{-8}$} (3.west)
+            (2.east) edge [bend left] node[right] {$\fib{-8}$} (3.east)
+            (3.west) edge [bend right] node[left]{$\div \fib{5}$} (4.west)
+            (3.east) edge [bend left] node[right] {$\div \fib{5}$} (4.east);
+    \end{tikzpicture}
+\end{figure}
+
+Check:
+
+$7\fib{x} + 8 = \fib{-14} + 8 = \fib{-4}$\\
+$2\fib{x} - 2 = \fib{-6} - 2 = \fib{-4}$
+
+\newpage
+
+\question[3]
+
+Susie clears out boxes on the weekend. She gets paid £10 for the weekend and an
+extra £2 for each box she clears. If Susie is paid £40, how many boxes did she
+clear? \droppoints
+
+\begin{minipage}{0.69\linewidth}
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+        thick,main node/.style={draw=none,fill=none}]
+
+        \node[main node] (1) {$\fib{10 + 2x} = 40$};
+        \node[main node] (2) [below of=1] {$\fib{2x} = \fib{30}$};
+        \node[main node] (3) [below of=2] {$\fib{x} = \fib{15}$};
+
+        \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$-\fib{10}$} (2.west)
+            (1.east) edge [bend left] node[right] {$-\fib{10}$} (2.east)
+            (2.west) edge [bend right] node[left]{$\div\fib{2}$} (3.west)
+            (2.east) edge [bend left] node[right] {$\div\fib{2}$} (3.east);
+    \end{tikzpicture}
+\end{figure}
+\end{minipage}
+\begin{boxedminipage}{0.3\linewidth}
+    LOOK!:
+
+    This question is asking you to make the linear equation yourself.
+
+    HINT:
+
+    What is the unknown?
+\end{boxedminipage}
+
+Check:
+
+$\fib{2}\fib{x} + \fib{10} = \fib{30} + \fib{10} = \fib{40}$
+
+\newpage
+
+\question[3]
+
+Rearrange to find the unknown:\\
+$3z - 18 = 7z - 10$ \droppoints
+
+\begin{minipage}{0.69\linewidth}
+\begin{figure}[H]
+    \centering
+    \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.5cm,
+        thick,main node/.style={draw=none,fill=none}]
+
+        \node[main node] (1) {$3z - 18 = 7z - 10$};
+        \node[main node] (2) [below of=1] {$\fib{3z} = \fib{7z + 8}$};
+        \node[main node] (3) [below of=2] {$\fib{-4z} = \fib{8}$};
+        \node[main node] (4) [below of=3] {$\fib{z} = \fib{-2}$};
+
+
+        \path[every node/.style={font=\sffamily\small}]
+            (1.west) edge [bend right] node[left]{$\fib{+ 18}$} (2.west)
+            (1.east) edge [bend left] node[right] {$\fib{+ 18}$} (2.east)
+            (2.west) edge [bend right] node[left]{$\fib{- 7z}$} (3.west)
+            (2.east) edge [bend left] node[right] {$\fib{- 7z}$} (3.east)
+            (3.west) edge [bend right] node[left]{$\fib{\div -2}$} (4.west)
+            (3.east) edge [bend left] node[right] {$\fib{\div -2}$} (4.east);
+    \end{tikzpicture}
+\end{figure}
+\end{minipage}
+\begin{boxedminipage}{0.3\linewidth}
+    NOTE:
+
+    This question uses z as the unknown. Any letter may be used as the
+    unknown.
+
+\end{boxedminipage}
+
+Check:
+
+$3\fib{z} - 18 = \fib{-6} - 18 = \fib{-24}$\\
+$7\fib{z} - 10 = \fib{-14} - 10 = \fib{-24}$
+
+\end{questions}
+
+\end{document}
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