TITLE: Can I think of Algebra like this?
QUESTION [102 upvotes]: This year in Algebra we first got introduced to the concept of equations with variables. Our teacher is doing a great job of teaching us how to do them, except for one thing:
He isn't telling us what we are actually doing when simplifying/solving for an equation.
Instead of telling us we are adding or subtracting something from both sides, he tells us we are just moving something over the equals sign.
Take, for example, this simple equation.
$3x+5=2x+10$
We have to get all "x" terms on one side, so I originally thought we subtract 2x from both sides, leaving
$x+5=10$
But that isn't how he teaches it.
He says:
We have to get all "x" terms on one side, so we move 2x over the equals sign, and whenever anything goes over the equals sign it becomes negative, so we have:
$-2x+3x+5=10$ and then we can combine like terms to get
$x+5=10$
Yes, they are basically doing the same thing, but my teachers way over complicates things a bit, and my main concern is that my classmates seem to think we can move the $2x$ over by "magic" and don't know that we are just subtracting it from both sides. 
This at first seemed really bad to me, but it seems in everything we have done so far, you can get away with not really knowing what you are doing while doing it. And, for some reason, thinking about doing Algebra this way seems to make it easier for me and my peers.
My question is:
Is there any disadvantages to thinking about Algebra like this? Is there anything later in my math education that will require me to know that I am subtracting or adding 2x to get rid of it on this side?

REPLY [7 votes]: This way of thinking about linear equations is very common in physics and engineering. I'll illustrate how it appears there by way of an example.
Suppose you're a plumber, looking at a junction where three pipes meet. Water flows into the junction through pipes 1, 2, 3 at rates $Q_1, Q_2, Q_3,$ respectively. Water can't be compressed or expanded (not as far as a humble plumber is concerned, anyway), so the net flow rate into the junction has to be zero:
$$\begin{align*}
\text{water in} & = 0 \\
Q_1 + Q_2 + Q_3 & = 0.
\end{align*}$$
This is a funny way to describe the junction, because water can't be flowing inward through all three pipes: at least one of the flow rates $Q_1, Q_2, Q_3$ has to be negative. Sometimes it's convenient to divide the pipes into "in-pipes" and "out-pipes," and rewrite the equation above as the statement that the flow in has to balance the flow out.
Suppose, for example, that we want to think of pipe 3 as an out-pipe, placing its flow on the other side of the balance sheet. The outward flow rate through pipe 3 is $-Q_3$, the negative of the inward flow rate $Q_3$, so the equation describing the junction becomes
$$\begin{align*}
\text{water in} & = \text{water out} \\
Q_1 + Q_2 & = -Q_3.
\end{align*}$$
We could just as easily designate both pipe 1 and pipe 3 as out-pipes, leaving pipe 2 as the sole in-pipe:
$$\begin{align*}
\text{water in} & = \text{water out} \\
Q_2 & = - Q_1 - Q_3.
\end{align*}$$
We could even call all three pipes out-pipes:
$$\begin{align*}
\text{water in} & = \text{water out} \\
0 & = - Q_1 - Q_2 - Q_3.
\end{align*}$$

All of these equations have the same physical meaning. From this point of view, what you're "actually doing" when you rewrite linear equations is describing the same physical situation from different points of view.

The "balanced junction" way of thinking can be applied to many situations. Instead of pipes carrying water, we could be looking at wires carrying current, or particles carrying momentum. Particle physicists find this way of doing algebra so useful that they have a special name for it: crossing symmetry.
For some purposes, the "balanced junction" point of view can be more convenient than the "matched operations" point of view that you prefer. For other purposes, like working with non-linear equations, the "matched operations" point of view has distinct advantages. The "matched operations" point of view may be more general, but I think it's a mistake to call it more fundamental.

What you're "actually doing" when you do algebra depends on your point of view—on the situation you're looking at, and the way you're looking at it.