Can I think of Algebra like this?

Question

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?

Answer 1

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?

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.

Answer 2

It's important to understand both points of view. In particular, if you ever need to ask yourself why it's okay to "move things around", you could always think about it and remember that all you're really doing is applying the same change to two sides of an equation.

What we have here is a common occurrence in mathematics: two different points of view of the same problem are each useful for different reasons. In the end, it is by synthesizing these points of view that we gain a deeper understanding of the problem at hand and grow as mathematicians.

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As for those who don't realize that you're "really" doing the same thing to two different sides: I wouldn't worry too much about it. In all likelihood, it will come up in a math problem one day and suddenly the light in their head will switch on and they'll realize "oh, those two different things that I've learned are really the same". I know that the same thing happens to me on a regular basis.

And for those who never have that epiphany: well, some people insist on remembering things the hard way, and go on to complain that math is a bunch of ad hoc and arbitrary rules. If you find a way to change that, do let me know.

Answer 3

One of the unfortunate things about math in school is that teachers are often tasked with getting students to understand one particular thing. So they'll do whatever they can to teach that one thing so the students can pass the test, even if it means that the students may not truly understand the underlying concept, making it harder for them to learn more advanced topics. (By that time, they'll have moved on to a different teacher...)

In this case, your teacher is tasked with getting you to understand addition and subtraction in algebra. This can be tough to understand, so he invents this cute story about things becoming negative when they move through the equals sign. Ok. That will work for solving stuff like $x+5 = 10$.

But what about equations like $6x = 18$? At this point, you have to multiply or divide, where the cute story about moving stuff through the equals sign falls flat on its face and your teacher will have to make up a new cute story. Eventually, all of these cute stories will accumulate to create profound confusion, and after years of studying algebra, the students will still not understand the underlying concepts.

I'll try to set the record straight. Algebra is (usually) about doing two things:

1. Maintaining equality
2. Solving for an unknown

You already know about the solving for an unknown part -- you're always trying to solve for $x$. That's the goal.

The first part is a little trickier to understand. We usually have two or more expressions (for example, $x+5$ and $10$) that are connected by an equals sign. This means that these two expressions are equal. The numbers they represent are the same. What we then do is manipulate these expressions, while maintaining their equality, to achieve our goal: solving for the unknown.

So when we have a statement like $5x = 10+2x$, we want to manipulate the expression on the left and the expression on the right, and keep them equal at all times. If our manipulation is skilful, then we'll eventually end up with just one $x$ on one side, and we'll have found out what it is equal to.

So let's take a statement like $a = b$. Fundamentally, maintaining equality means that whatever you do to $a$, you must do to $b$. It makes sense, right? If $a$ and $b$ are equal, then if you add $5$ to $a$ and want to keep $a$ equal to $b$, you must also add $5$ to $b$. If you multiply $a$ by $10$, you must also multiply $b$ by $10$. If you square $a$, you must square $b$. So on, so forth.

But there are caveats. What do we really mean when we talk about equality in this context? Take an equation like $x^2 = 25$. Either $x=5$ or $x=-5$ will be a valid solution: both of those values for $x$ will let the statement of $x^2=25$ be true. So when we solve $x^2=25$, we can't just take $\sqrt{25}=5$ and say we're done. We may have done the same thing to both expressions, but we did not maintain equality.

Or take a statement like $x=1$. We can square both sides and get $x^2 = 1^2 = 1$. Now, $x^2 = 1$ has solutions $x=1$ and $x=-1$. Again, we did the same thing to both sides, but we did not maintain equality. So what does it mean to maintain equality?

We start with a statement that is true, like $x^2 = 25$. The unknown in that statement is $x$. There is a set of solutions (that is, a bunch of numbers) for $x$ that will let $x^2=25$ be true. $5$ and $-5$ are both in that set. $6$ is not in that set because $6^2 = 36$, and $36 \neq 25$.

In the $x=1, x^2=1$ example, we did not maintain equality because we changed the solution set. There's only one value for $x$ that makes $x=1$ true, and that's $1$. But there are two values for $x$ that make $x^2=1$ true: those are $1$ and $-1$. Let's call those the potential solution set.

So we have to go back to our original equation, the one we know to be true, and check the values we got in the potential solution set. In this example, our equation was $x=1$. So we pick the first number from the potential solution set: $1$. We plug it in for $x$. $1=1$. Great. Now we try the other number: $-1$. Plug it in for $x$. $-1 = 1$. Nope.

So to summarize, we maintain equality between two expressions by doing two things:

1. Doing the same things to both expressions.
2. Watching out for operations that can change the solution set to an equation (these are most commonly squaring/square roots, i.e. exponentiation, or trigonometric operations, like $\sin$, etc.). When dealing with those kinds of operations, it's a good idea to take the potential solutions and to plug them into your original equation to see which ones work and which ones don't.

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I would strongly encourage you to always solve these problems with the mindset of maintaining equality, rather than by using the stories your teacher may tell. Beyond what I've said above, there are two big reasons for this:

1. By thinking about equality, you will think more about the underlying mathematics, rather than just about moving around symbols.
2. The problems you will encounter in Algebra (and in math in general) will gradually become harder, so to be successful, you'll need to actually understand the underlying notions.

Anyway, let's solve the problem $3x + 5 = 2x + 10$ with the kind of mindset I've advocated. For an easy problem like this, it may be overkill, but it'll certainly be useful to think in these terms when you're facing harder problems. $$3x+5 = 2x+10$$ These statements are equal. Subtract $2x$ from each expression. $$3x-2x + 5 = 2x - 2x + 10$$ Simplify. $$x + 5 = 10$$ Subtract $5$ from each expression. $$x + 5 - 5 = 10 -5$$ Simplify. $$x = 5$$

Answer 4

I've been angry at my math teachers for years for not telling me why stuff behave the way they do, but teaching me bunch of algorithms instead. That is, until I started teaching mathematics in elementary school after finishing pure maths.

You are understanding correctly what is actually going on. This is fantastic, and I'd be proud to teach someone like yourself. But please understand that not many students will find this more clarifying than "moving stuff around". Unfortunately, teachers need to comform to majority most of the time, so they are often forced to take more simple approach compared to more rigorous one. Try not to be too critical of your teacher and instead raise good questions, like this one, in your class. Any good teacher will be happy to discuss different points of view and that will be beneficial for your whole class.

Answer 5

I agree with you. There is no theorem or postulate about "moving sides and changing signs." There are rules, however, that allow you to add (or subtract) the same quantity, whether a constant or variable from both sides of the equal sign.

In my math center (a tutoring room in the high school where I work), I see many students who will subtract the same number from both sides, writing it our each time. At some point in their progress, I'd hope they can see $X+3=10$ and know that $X=7$ is the next step without laboriously writing "$-3$" under both sides of the equation. It seems your teacher isn't just encouraging you all to skip a step or make this small leap, but rather ignore the rules altogether. Not really the best pedagogical process in my opinion.

Answer 6

Your teacher is right because he is describing things on a higher level.

You can indeed add terms on both sides of an equation. Like

$$3x+5=2x+10$$

which can become

$$9x+3x+5=9x+2x+10$$

or $$12x+5=11x+10.$$

But what he means is that he is adding the exact term necessary to let the unknown disappear from a member.

$$-2x+3x+5=-2x+2x+10$$

or $$x+5=10.$$

This way, he is indeed "moving" all $x$'s from the RHS to the LHS to simplify the equation. This expresses the purpose rather than the means.

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There are other jargon expressions which come handy when you explain a computation.

For instance "completing the square".

To solve the equation

$$x^2+6x+5=0,$$ you add $4$ to both members to get

$$x^2+6x+9=4.$$

Then you have a perfect square and you can solve more easily

$$(x+3)^2=2^2.$$

Once again you added terms to both sides, but you say "complete the square".

Answer 7

Is there any disadvantages to thinking about Algebra like this?

The disadvantage of thinking about algebra as 'moving things around' is that there is no justification that moving the 2x over to the other side of the equation $\color{#180}{\mathsf{will} \space \mathsf{cause}\space \mathsf{the}\space \mathsf{sign}\space \mathsf{of} \space 2x \space \mathsf{to} \space \mathsf{change}}$.

We know of course that it will have to change sign as equations really do behave as if they were 'weighing scales'; operations are carried out on both sides (to keep it balanced).

I strongly think you should take a look at this, the user @Aaron has posted some excellent comments with arguments that back up what I am saying here.

So an interesting question would be to ask your teacher why the $2x$ changes sign?

I'm thinking that at this point your teacher may have no choice but to give you the weighing scales analogy.

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?

Yes, take functions as an example. The Cosine function or inverse Cosine function must be applied to both sides of the equation.

Now this is where I get to make my point; imagine you were asked so solve $\cos\theta=\frac13$. You know intuitively that to "get rid of" the $\cos$ in front of the $\theta$ you write (assuming $0\le \theta \le \pi$):

$$\cos\theta=\frac13\tag{A}$$ $$\implies\theta=\cos^{-1}\frac13$$

Notice that in $(\mathrm{A})$ nothing got 'moved', we didn't move the $\cos \theta$ to the other-side and switch its sign. Or put in a different way there was no $\cos$ to move over: there was only a $\cos \theta$ instead.

Assuming $0\le \theta \le \pi$ what we did do is :

$$\cos\theta=\frac13$$ $$\implies\color{blue}{\cos^{-1}} \cos\theta= \color{blue}{\cos^{-1}}\frac{1}{3} \tag{*}$$ $$\implies\theta=\cos^{-1}\frac13$$

Where in $\mathrm(*)$ we took the inverse cosine or $\arccos$ of $\color{red}{\mathrm{both}}$ sides: $\cos$ didn't move as it was never there to begin with.

So this is why you must think of algebra the way you were before your teacher said things differently. But also acknowledge that sometimes people explain things loosely; obviously, your teacher knows what's really going on but it takes less words to say "move the $2x$ over" instead of "add $2x$ to both sides of the equation".

As long as you know the truth in the back of your mind that's all that matters.

Answer 8

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.

Answer 9

I learned algebra using the "moving" procedure. My teacher explained me the fundamentals, but also explained me the "moving trick".

I always had clear that this actually was a "simplification" or "short-cut" of adding / subtracting / multiplying / dividing both sides of the equation. But I loved to do the "moving trick" as it was way more fast to solve equations. And in an exam, time is of the essence. Thanks to that, I always was an "A" student in math.

So my opinion is that makes no harm on doing tricks to accelerate procedure, provided that you know exactly what is the foundation of the trick and it's no "magic" at all.

And yes, you need to have clear the limitations - no dividing by zero, etc.

Answer 10

Is there any disadvantages to thinking about Algebra like this?

I believe there's none. "Moving" is really convinient trick to avoid it due to some "puristic" reasons. After all, the notion of "subtraction" is an overhead too, because it's no more than addition of negative number. Yet we still learn in the school 4 arithmetical operations, while only 2 are enough.

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?

Well, one should always remember that by "free moving" equation terms he's going to exploit associativity and (probably) commutativity properties. Dealing with non-commutative operations using "moving" abstraction is error-prone.

Answer 11

I think it is really the same if you stick to addition and subtraction.

With $3x+5=2x+10$ "moving the $2x$" is really just subtracting $2x$ from both side but combining the moves. Where the distinction becomes important is dividing. i.e. if you had $2(a-b)x=10(a-b)$ it is OK to cross out the $(a-b)$ on each side. But remember you are really dividing. So it is OK unless $a-b=0$.

Answer 12

What's going on here is difficult to explain without the concept of a function, so I'll assume this concept is already understood. It turns out that both methods are equally valid; the reason has to do with the definition of inverse functions. The story goes something like this:

Consider functions $g : X \leftarrow Y$ and $f : Y \leftarrow X$.

Proposition. Then the following are equivalent.

1. For all $x \in X$, $g(f(x)) =x$, and for all $y \in Y$, $f(g(y))=y$.
2. For all $x \in X$ and $y \in Y$, we have $g(y) = x \iff y = f(x).$

Definition. If either (and hence both) of the above conditions hold, we say that $g$ and $f$ are inverse functions.

Returning to your question:

The relevant functions are $(+2x)$ and $(-2x).$

1. Your method, whereby you apply the same function to both sides, makes use of Condition 1. It first applies $(-2x)$ to both sides, and then uses the fact that this is the inverse of $(+2x)$ to simplify the expression on the right-hand side.

2. Your teacher's method, whereby you "move things over the equals sign," makes use of Condition 2. It simply moves the $(+2x)$ function from one side to the other, and it becomes $(-2x)$ in the process, in line with Condition 2.

Anyway, both methods are equally valid.

Now you ask:

Are there any disadvantages to thinking about algebra like this?

No. The more perspectives, the better. Just try to be flexible.

Answer 13

In a v.simple 1st degree equation : one unknown, such as given, subtracting from both sides and "moving" a term are EQUIVALENT.Moving a term to the left amounts to subtracting from both sides, but abbreviating by omitting the subtraction notation on R.H.S...abbreviation is the Fundamental Confusion of mathematics. Yes, for further studies you will need to know that equation- solving techniques depend on reductions to lower degree and are consequences of Field Axioms.