Principles for Solving an Equation

Questions about solving algebraic equations are common. Here I will bring together several answers where we discussed the basic principles for solving relatively simple equations, which are important to learn well before moving on to quadratic equations and beyond.

Two principles

First, we’ll start with a 2001 question about a simple linear equation:

Solve 2x + 3 = 10

My problem: 2x + 3 = 10

You are supposed to subtract 3 from the 10, which is 7, then divide by two, which is 3.5x + 10. I don't get what to do next to simplify the problem.

Billy clearly is just beginning to learn algebra; it is common to use wrong terms (“simplify the problem” rather than “solve the equation”) and to misunderstand what one is told to do. This was a case for going through the whole problem step by step, with an emphasis on the underlying reasoning, as Doctor Rick did:

The goal is to *solve* the equation; that is, to find the value of x for which the equation is true. There are two basic principles that we use to do this. 

One I call the principle of "undoing." Look at the expression on the left, 2x+3. According to the order of operations, it is built out of just-plain-x in two steps: first multiply the x by 2, then add 3. We want to get just-plain-x alone on one side of the equation. To do this, we "undo" the steps by which the expression was built.

When you get dressed, first you put on your socks, then you put on your shoes. When you get undressed, first you take off your shoes (the *last* thing you put on), then you take off your socks. So here: we undo the last thing first. That's why you first subtract 3, which undoes the addition of 3. Then you divide by 2, which undoes the multiplication by 2.

In explaining this concept, I often have to urge students first to focus on what operations are involved, just as Doctor Rick did here. Multiplications can be almost invisible! Once you see the operations, you know to undo them, and the appropriate order. (In more complicated equations, it is best to simplify before we start this solving process, so that there are fewer operations to juggle; but sometimes we don’t, which can change the order. And, as I have said elsewhere, it is possible to take off your socks before your shoes – you just end up with stretched socks and pinched fingers. All rules can be broken, if you know what you are doing.)

The other principle is "keeping the balance." The equation is like a scale with two pans that balance. They balance because the two pans contain the same weight. In the same way, the two sides of the equation balance because have the same value - they equal the same number. (We don't know what that number is yet.)

We want to *keep* the equation balanced. If we take a weight off one pan of a balance scale, we must take the same amount off the other side, or it will no longer be balanced. In the same way, if we do something to one side of the equation, we must do the same to the other side, otherwise it will go out of balance. So, when we subtract 3 from one side, we must subtract 3 from the other side, too. When we divide one side by 2, we must divide the other side by 2 also.

This rule is often misused; I’ll get to that below. A key idea is that the “something” that you do to each side must be something you do to the entire value of each side (like dividing the whole thing by 2), not something piecemeal like squaring each term, or dividing only one term by 2.

Solving the equation

Now we have a plan (subtract, then divide), and a procedure (do that to both sides at once). Let’s do it:

Now I'll show you how these principles work to solve your equation.

  2x + 3 = 10

First we subtract 3 from *both* sides. Then we have

  2x = 10 - 3

Doing the subtraction on the right, we have

  2x = 7

Then we divide *both* sides by 2. Then we have

  x = 7/2

That's the solution: if x is 7/2 then the equation is true. Let's check this out: I will replace x by 7/2 in the equation.

  2(7/2) + 3 = 10

  7 + 3 = 10

  10 = 10

Both sides have the same value, so the equation balances. If we chose a different value for x, then the final equation, x = 7/2, wouldn't balance, and so the original equation, 2x + 3 = 10, wouldn't balance, either: 2x + 3 would not be equal to 10.

Interestingly, Doctor Rick’s two principles are essentially what gave algebra its name. I don’t know Arabic well enough to get this right, but the name of the original book about algebra, by the Persian mathematician al-Khwarizmi, was Kitab al-Jabr wa-l-Muqabala (see Wikipedia), which is translated in various ways, such as “Completion and Balancing”. “Completion” (or “forcing” or “restoring”), as I understand it, was essentially “undoing”, and “balancing” was “keeping the balance”. It’s not exactly the same, but close enough.

Applying a function to each side

The balancing principle is often stated in more specific ways; for example, a textbook might state an Addition Property of Equality (that an equation remains true if you add the same thing to both sides) and a Multiplication Property of Equality (that an equation is still true if you multiply both sides by the same quantity). I find that such terms can confuse students in a couple ways, as shown in the next answer, from 2008.

Equality Properties and What They Really Mean

In class we are shown how to square both sides of an equation or take the square root of both sides of an equation but is there a rule like the addition property of equality or multiplication property of equality that says it is ok to do so?

I have asked the instructor, looked at algebra text books and searched Dr. Math.

Some textbooks state such a rule; but do we need a separate rule for everything you will ever want to do to both sides? Not really, as I answered:

Interesting question!  I think it shows that these properties really shouldn't be taught in this way, which makes things simple for teaching kids but doesn't accurately reflect what is actually happening.  The so-called addition and multiplication (and subtraction and division) properties of equality are not really properties of equality in the first place, but are facts about each operation.

Listing properties makes them easier to talk about at first, but only complicates things later on. Since Margaret was beyond the beginning, I gave her a more mature explanation.

I believe you are talking about facts like this, the multiplication property of equality:

  If a = b and c is any real number, then ac = bc.

The idea is that if two numbers are really the same number, then when we multiply them both by the same thing, we get the same answer.  How could we not?  As long as multiplication is "well-defined"--that is, always gives the same answer--this has to be true.  The same is true of any other operation, including powers, square roots, reciprocals, and so on!  Any well-defined operation (or function, in fact) will behave this way.  The only thing that could go wrong, really, is if you can't perform the operation at all (e.g. if you want to take the square root of both sides but one or both may be negative).  This becomes a domain issue, if you are familiar with functions.

So you don't really have to look for specific properties of equality associated with each operation you want to use; you just have to determine that it is well-defined (has one value) and that its domain includes the values to which it is being applied.  These two facts amount to the property you are looking for.

Knowing this, for example, we don’t need a new property if we want, say, to take the logarithm of both sides of an equation. We just need to know that the logarithm of any given number is always the same (so that the result, if two expressions are equal, is two logs that are equal), and to be aware that we have to check that our solution is not invalid (extraneous).

Note, again, that this is what we mean by “doing the same to both sides”: applying the same function to the two equal expressions in the equation. I imagine the main reason for using terms like “addition property of equality” is that beginning students don’t yet have words like “function” to clarify what “doing” means.

The importance of equivalent equations

But there’s more: It is not enough to know that the new equation is true whenever the original equation is true; that only guarantees that we will find all the solutions. What if the new equation is true in some cases where the original is false? Then we will find “solutions” that are not – they will be extraneous. We need something more:

Now, some texts define the "X property of equality" as something a bit deeper than what I have just discussed:

  If c is any real number other than 0,
  then the equation ac = bc is EQUIVALENT
  to the equation a = b.

This is what you REALLY need to use when you solve an equation; it says not only that ac is still equal to bc, but that they are ONLY equal if a = b; you don't either lose or gain solutions.

This property is not necessarily true for any well-defined operation (or for any function), but for any INVERTIBLE (one-to-one) function (that is, when there is only one way to get any given result).  In particular, it is NOT true for even powers, because there are two different numbers with the same square, so that for example 1 = -1 is not true yet (1)^2 = (-1)^2 is true.  They are not equivalent.  This is why squaring both sides of an equation can yield extraneous solutions.

So, taking the logarithm of both sides is fine, because it is one-to-one (it is its domain that causes extraneous solutions); but squaring causes trouble by not being one-to-one. We just have to live with that, remembering to check. Similarly, we can’t multiply or divide by a variable quantity (that might be zero), because division by zero is not defined, and multiplication by zero is not one-to-one.

A related issue comes up with square roots.  Although it is true that if a = b, then sqrt(a) = sqrt(b), and in fact these equations are equivalent if you ignore domain issues, this can lead to problems when you forget that the radical symbol means only the positive root, and try to apply it to something like this:

  (-1)^2 = 1^2

Taking the square root of each side by just canceling out the squares, you'd get

  -1 = 1

which is not true!  You might not do this here, but you probably would if there were variables:

  x^2 = y^2

does not imply that

  x = y

because one might be positive and the other negative!  What's happening here is that, on one hand, you are unconsciously using a form of the square root that is not a function (that is, has more than one value) by allowing a negative result; or, on the other hand, you are forgetting that in reality

  sqrt(x^2) = |x|, not x.

So squaring and square roots both come with warning labels! If you square both sides, you have to check for extraneous solutions, and if you take a square root, you have to remember the “±” indicating two possible roots.

So the answer to your question really depends on exactly how your "multiplication property" and so on are defined, and what you want to use your new property to do.  Perhaps if you show me how you want to use it, I can clarify what I am saying.  Hopefully in taking squares or square roots in equations you have been taught the caveats that arise; some books may present these facts as properties, including all the warnings, but others pass by them all too quietly!

Avoiding lost solutions

Some closely related ideas are discussed in this answer from 2011:

Algebra for Equivalence

My teacher explained in class that I would lose possible solutions to trigonometric equations if I divided by a function or took the square root of a function.

I want to avoid any kind of loss. What are all the operations that do this in trigonometry?

And could you please explain to me how they destroy solutions?

This question is really about algebra; I didn’t refer to trigonometry at all in my answer:

In general (and this has nothing to do with trig, really), you should avoid doing anything that you don't KNOW will produce an equivalent equation. Don't just do things that feel reasonable; make a list of what has been proved to be valid in this context. When in doubt, don't; or at least, go check it out and make sure you can do it.

The examples you give (which may be the only ones at your level) are the opposite of the actions that produce extraneous solutions -- but these ideas are closely related. The actions you're asking about accidentally reduce the solution set of a problem so that you miss solutions; the opposite actions accidentally expand the solution set so that you find solutions that are not solutions of the original.

So the main thing to do is proactive: only do what you know will not lose solutions. Then you don’t need a list of what to avoid, just a list of what to do. But it’s good to understand why this happens:

Don’t divide by a variable

First, if you divide both sides of an equation by an expression containing a variable, you are making an assumption: that what you are dividing by is not zero. If, for some value of the variable, the divisor is zero, then since division by zero is not defined, that value may not be a solution of the new equation, even though it was a solution of the original. It's as if you were searching for a criminal hiding in your neighborhood, and you don't bother to look in your own house because that possibility doesn't even occur to you. If he's there, it's your own fault that you didn't find him.

For example, take the simple equation

   x^2 = x

If you see that you can divide both sides by x, and do so, you get

   x = 1

But in dividing by x, you were assuming that x was not zero. If x were zero, then when you had ...

   x^2    x
   --- = ---
    x     x

... you would really have 0/0 on both sides; the left side is not x, and the right side is not 1. This case is just quietly swept under the rug when you simplify.

So you really have to make a separate check for that case. Here, it turns out that x = 0 is in fact a solution. You missed it because you didn't consider the possibility.

Division by something that might be zero is the opposite of multiplying by something that might be zero: the former can lose solutions, the latter can gain extraneous solutions.

But there’s a way around this:

The better way to do this, which doesn't require a separate check, is to factor rather than divide:

   x^2 - x = 0

   x(x - 1) = 0

   x = 0 or x - 1 = 0

   x = 0 or x = 1

This will always work if you could have divided; it takes that x you divided by in the other method, and keeps it out in the open, so you won't miss it.

So, factoring is the safe alternative when you are tempted to divide.

Be careful with square roots

As for taking a square root, I'd like to see an example where you've done that, but the case I have in mind is like this:

   x^2 = 1

   sqrt(x^2) = sqrt(1)

   x = 1

Here, the issue is a little different: nothing is undefined; you just forgot that any positive number has TWO square roots, not just one. The radical symbol represents only the principal root (the non-negative one), so you have to explicitly put a plus-or-minus symbol before it to tell you to use both signs:

   x^2 = 1

   sqrt(x^2) = +/-sqrt(1)

   x = +/-1

   x = 1 or x = -1

(By the way, technically we should put the +/- on the left side as well, but that wouldn't gain us anything, since changing the sign on one side has the same effect as changing the sign on the other.)

A mistake like this was at the root (pardon the pun) of a recently discussed error.