Combining arithmetic types

Next: Summary: The Rules of Up: Movement As Arithmetic Previous: Multiplication/Division Class

Combining arithmetic types

Here is a more complex example, which is solved with a combination of the above methods. First we will briefly solve it using the normal algebraic methods. Then we'll solve it again using symbol motion. In this equation,

$\begin{displaymath}\frac{3x+7}{2x+1}=\frac{5}{2}\end{displaymath}$

one way to solve it is to first multiply each side by 2:

$\begin{displaymath}2\times\left(\frac{3x+7}{2x+1}\right)=2\times\left(\frac{5}{2}\right)\end{displaymath}$

$\begin{displaymath}\frac{2\times\left(3x+7\right)}{2x+1}=5\end{displaymath}$

Then, multiply each side by (2x + 1).

$\begin{displaymath}(2x+1)\times\left(\frac{2\times\left(3x+7\right)}{2x+1}\right)=(2x+1)\times5\end{displaymath}$

$\begin{displaymath}2\times\left(3x+7\right)=(2x+1)\times5\end{displaymath}$

$\begin{displaymath}6x+14=10x+5\end{displaymath}$

Collect the variables on one side and the numbers on the other, and simplify.

$\begin{displaymath}6x+14-6x=10x+5-6x\end{displaymath}$

$\begin{displaymath}14=4x+5\end{displaymath}$

$\begin{displaymath}14-5=4x+5-5\end{displaymath}$

$\begin{displaymath}9=4x\end{displaymath}$

$\begin{displaymath}x=\frac{9}{4}\end{displaymath}$

Now, how do we do the same thing through symbol motion? Before we start, it's good to point out a few things. First, when moving symbols, we obey the normal rules of arithmetic. For example, if we are solving this equation:

$\begin{displaymath}\frac{2}{x+3}=1\end{displaymath}$

we would chunk the x + 3

$\includegraphics{figures/movsym/fig_movsym_ka.eps}$

and begin to move it to the right side.

$\includegraphics{figures/movsym/fig_movsym_kb.eps}$

We would certainly NOT do something like this:

$\includegraphics{figures/movsym/fig_movsym_kc.eps}$

We can always be clear by asking what sequence of arithmetic operations that the symbol motion is supposed to emulate. Here is a general principle:

Every symbol motion has the same effect as a sequence of mathematical operations.

If (and only if) some change in the image is equivalent to some such sequence, is that change a valid symbol motion. If there is no such sequence of operations, the change is not a symbol motion and has no mathematical meaning.

Let's work the equation at the beginning of this section again, this time using symbol motion.

$\includegraphics{figures/movsym/fig_movsym_la.eps}$

We'll start by making two chunks, both of which will be moved:

$\includegraphics{figures/movsym/fig_movsym_lb.eps}$

Both of these chunks are in the denominators. We move each of them to the numerator of the opposite side:

$\includegraphics{figures/movsym/fig_movsym_lc.eps}$

As we do this, we erase the horizontal lines, and insert multiplication symbols ( $\times$ ) as appropriate.

$\includegraphics{figures/movsym/fig_movsym_ld.eps}$

Notice that we have also put parentheses around two of the expressions. This is sometimes needed when symbols are moved around, for the same reason they are needed in normal algebraic manipulation. Next, we distribute the multiplication on each side:

$\includegraphics{figures/movsym/fig_movsym_le.eps}$

Chunk the 6x and the 5, and begin to move them.

$\includegraphics{figures/movsym/fig_movsym_lf.eps}$

As we do this, we switch the signs of each, from 6x to -6x and 5 to -5.

$\includegraphics{figures/movsym/fig_movsym_lg.eps}$

$\includegraphics{figures/movsym/fig_movsym_lh.eps}$

Normal arithmetic quickly solves the rest ( $x=\frac{9}{4})$ .

Next: Summary: The Rules of Up: Movement As Arithmetic Previous: Multiplication/Division Class