# Can I think of Algebra like this?

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?

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?

As a college algebra instructor, I’m involved with remediation efforts for hundreds of students each year who have graduated high school but can’t get started with college math, mostly due to incorrect concepts picked up in their prior schooling. So I would say “yes”. There are some shortcuts that teachers can take to get students to pass some specific tests or programs that they are involved in; but the incorrect concepts definitely make things more difficult for students, sometimes overwhelmingly so, later on. (A majority of students that land in college remediation programs never get college degrees.)

The first thing that I would point out is that the “apply inverse operations to both sides” idea is generalizable to any mathematical operation; this allows you to cancel additions, subtractions, multiplications, divisions, exponents, radicals… even exponential, logarithmic, and trigonometric functions. (With appropriate fine print: no division by zero, square roots to both sides creates two plus-or-minus solutions, trigonometric inverses creates infinite cyclic solutions, etc.)

In contrast, the “move over and change the sign” method is not generalizable, as it only works for addend terms. This sets students on a course that requires memorizing many apparently different rules, one for each operation, which is much harder. When solving $2x = 10$, how is the multiplier of 2 canceled out? Must we remember to move it and turn it into the reciprocal 1/2? Will the students mistakenly change the sign and multiply by -1/2? Or add or multiply by -2 (I see this a lot)? How do we remove the division in $\frac{x}{2} = 5$ (probably some other rule)? How will we remember the seemingly totally different rule to solve $x^2 = 25$?

By way of analogy, I have college students who never memorized the times tables; they did manage to get through high school by repeatedly adding on their fingers, and can get through perhaps the first part of an algebra course that way. But then we start factoring and reducing radicals: “What times what gives you 54?” I might ask; “I have no idea!” will be the answer (this happened this past week; and here’s a student who has effectively no chance of passing the rest of the course).

In summary: There are shortcuts or “tricks” that can get a student through a particular exam or test, which prove to be detrimental later on, as the “trick” fails in a broader context (like in this case, with any operations other than addition or subtraction). This then sets a student on a road to memorizing hundreds of little abstract rules, instead of a few simple big ideas, and at some point that complicated ad-hoc structure comes crashing down. Be polite and don’t fight with your teacher to change things; but make sure to pick up a broader perspective for yourself, and share it with other students if they’re willing, because you will need it later on. Take the opportunity to think about how you could improve on teaching the material, and then you may be on the path to being a master teacher yourself someday, and helping lots of people who need it.