Often times algebraic concepts have a geometric meaning too. You've now done enough arithmetic with pronumerals to be able to learn this secret of the universe.

Consider \(3(2x+5)\). You can expand this by distributing the \(3\) to each term in the brackets like so: 

\vspace{0.5cm}
\begin{equation} (\source{3}\source{})(\target{2x}+\target{5})=6x + 15\mbox{\drawarrowssingle} \end{equation}
\vspace{0.5cm}

Or you can think about this as having some kind of original rectangle with dimensions $ 3 $ by $ 2x $ and then extending the width by $ 5 $.

\begin{center}
    \begin{tikzpicture}[scale=0.5]
        \draw (0,0) rectangle (7,3);
        \draw (2,0) rectangle (7,3);

        \node at (1,3) [above] {$2x$};
        \node at (0, 1.5) [left] {$3$};
        \node at (3.5, 3) [above] {$5$};
    \end{tikzpicture}
\end{center}

Now finding the area of the enlarged shape is algebraically equivalent to $ 3(2x + 5) $ and often times expanding this will make substitution easier if you know what the value of $ x $ is!

These kinds of expansions are the backbone of mathematics and becoming proficient at these will help you simplify harder problems. Let's get better at expanding:

\begin{examplebox}
    \subsection{Examples}
    \begin{enumerate}
        \item Expand:
            \begin{multicols}{3}
            \begin{enumerate}
                \item \(2(a+3)=\)\dotfill
                \item \(3(x-2)=\)\dotfill
                \item \(4(2m-7)=\)\dotfill
            \end{enumerate}
            \end{multicols}
        \item Now try:
            \begin{multicols}{3}
            \begin{enumerate}
                \item \(5(a+1)+6=\)\dotfill
                \item \(4(2b-1)+7=\)\dotfill
                \item \(6(d+5)-3d=\)\dotfill
            \end{enumerate}
            \end{multicols}
        \item Can you handle some more terms?
            \begin{multicols}{2}
            \begin{enumerate}
                \item \(2(b+5)+3(b+2)=\)\dotfill
                \item \(3(x-2)-2(x+1)=\)\dotfill
            \end{enumerate}
            \end{multicols}
    \end{enumerate}
\end{examplebox}


\begin{exercisebox}
    \subsection{Exercises}
    Have a go at these ones yourselves:
    \begin{multicols}{2}
    \begin{enumerate}
        \item \(\frac{3}{5}(6x+\frac{7}{3})=\)\dotfill
        \item \(\frac{4}{3}(6x+11)+\frac{2}{3}=\)\dotfill
        \item \(-12(4y-5)=\)\dotfill
        \item \(\frac{2}{3}(12p+6)=\)\dotfill
        \item \(-\frac{1}{2}(10d-6)=\)\dotfill
        \item \(-\frac{4}{5}(25m-100)=\)\dotfill
        \item \(\frac{3}{5}(\frac{x}{6}+\frac{1}{3})=\)\dotfill
        \item \(-\frac{3}{5}(\frac{a}{3}-\frac{2}{3})=\)\dotfill
        \item \(c(c-5)=\)\dotfill
        \item \(2i(5i+7)=\)\dotfill
    \end{enumerate}
    \end{multicols}
\end{exercisebox}
