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NSW Y8 Maths

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NSW Y8 Maths Graphing Linear Equations Solving Equations Graphically

Resources for Solving Equations Graphically

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Solving Equations Graphically Theory

\begin{multicols}{2} \textbf{Example 1}\\ By graphing the equation \(y=2 x+1\). Find the value of \(y\) when \(x=-1\).\\ \textbf{Example 1 solution}\\ \(y=2 x+1\)\\ when \(x=0, y=1\)\\ \(x=1, y=3\)\\ \(\therefore\) Plot and join the points \((0,1)\) and \((1,3)\).\\ Draw the line \(x=-1\)\\ The point of intersection is \((-1,-1)\)\\ \(\therefore y=-1\) \begin{center} \begin{tikzpicture}[scale=0.9] \def \xmax{3} \def \xmin{-3} \def \ymax{4} \def \ymin{-3} \def \xlabel{x} \def \ylabel{y} \begin{axis}[ axis lines=middle, axis line style={shorten >=-10pt, shorten <=-10pt,very thick,Stealth-Stealth}, grid=both, %major, major grid style={line width=1pt,draw=black!70}, minor x tick num=4, minor y tick num=4, ylabel = $y$, xlabel = $x$, width=3in, height=3in, ymin=\ymin, ymax=\ymax, xmin=\xmin, xmax=\xmax, axis on top=false, axis line style = thick, major tick style = thick, xtick distance = 1, xlabel style={at={(ticklabel* cs:1.05)},anchor=west}, x grid style={thin, opacity=0.5}, ytick distance = 1, ylabel style={at={(ticklabel* cs:1.05)},anchor=south}, y grid style={thin, opacity=0.5}, axis on top=false, xtick={-4,-3,-2,-1,0,1,2,...,4}, ytick={-4,-3,-2,-1,0,1,2,...,6}, nodes near coords style={ above right=3pt, inner sep=0, color=black, font=,}, ] \addplot[name path=a, ultra thick, latex-latex, samples=300, smooth, domain=-2:1.5, red] {2*\x+1} node [pos=0.9, left, red, font=\small] {}; \draw[blue,ultra thick,latex-latex] (-1,-3)--(-1,4); \addplot[black, mark=*] coordinates {(-1,-1)} node[left] {(-1,-1)}; \addplot[black, mark=*] coordinates {(0,1)} node[right] {(0,1)}; \addplot[black, mark=*] coordinates {(1,3)} node[below right] {(1,3)}; \end{axis} \end{tikzpicture} \end{center} \columnbreak \textbf{Example 2}\\ By graphing the equation \(1-3 x=7\), solve the equation for \(x\) graphically.\\ \textbf{Example 2 solution}\\ Let \(y=1-3 x\) and \(y=7\)\\ when\\ \(\begin{array}[t]{ll} x=0, & y=1 \\ x=-1, & y=4 \end{array}\)\\ plot and join the points \((0,1)\) and \((-1,4)\)\\ Draw the line \(y=7\)\\ The point of intersection is \((-2,7)\)\\ \(\therefore x=-2\). \begin{center} \begin{tikzpicture}[scale=0.9] \def \xmax{3} \def \xmin{-3} \def \ymax{8} \def \ymin{-2} \def \xlabel{x} \def \ylabel{y} \begin{axis}[ axis lines=middle, axis line style={shorten >=-10pt, shorten <=-10pt,very thick,Stealth-Stealth}, grid=both, %major, major grid style={line width=1pt,draw=black!70}, minor x tick num=4, minor y tick num=4, ylabel = $y$, xlabel = $x$, width=3in, height=3in, ymin=\ymin, ymax=\ymax, xmin=\xmin, xmax=\xmax, axis on top=false, axis line style = thick, major tick style = thick, xtick distance = 1, xlabel style={at={(ticklabel* cs:1.05)},anchor=west}, x grid style={thin, opacity=0.5}, ytick distance = 1, ylabel style={at={(ticklabel* cs:1.05)},anchor=south}, y grid style={thin, opacity=0.5}, axis on top=false, xtick={-4,-3,-2,-1,0,1,2,...,4}, ytick={-4,-3,-2,-1,0,1,2,...,8}, nodes near coords style={ above right=3pt, inner sep=0, color=black, font=,}, ] \addplot[name path=a, ultra thick, latex-latex, samples=300, smooth, domain=-2.3:1, red] {-3*\x+1} node [pos=0.9, left, red, font=\small] {}; \addplot[name path=b, ultra thick, latex-latex, samples=300, smooth, domain=-3:3, blue] {7} node [pos=0.9, left, red, font=\small] {}; \addplot[black, mark=*] coordinates {(0,1)} node[right] {(0,1)}; \addplot[black, mark=*] coordinates {(-2,7)} node[below left] {(-2,7)}; \addplot[black, mark=*] coordinates {(-1,4)} node[below left] {(-1,4)}; \end{axis} \end{tikzpicture} \end{center} \end{multicols}

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  • Solving Equations Graphically - Video - Graphs of linear equations | Linear equations and functions

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