Most students taking courses in Algebra or higher seem quite comfortable with the idea of “equivalent fractions”: improper or unsimplified fractions all of which evaluate to the same decimal value. An example would be
To create such fractions, multiply whatever fraction you wish to start with by 1 (the multiplicative identity) in the form of a fraction whose numerator and denominator are the same:
The key concepts here are that
a) an infinite number of equivalent fractions can easily be created, and
b) while all these equivalent fractions sure look different, they all represent the same decimal value or simplified fraction.
Turning to algebra, the very similar concept of “equivalent equations” is helpful in explaining how to solve algebra problems. I like to define “algebra” as: a set of rules for changing the appearance of an expression without changing the quantitative relationship that it defines. This is exactly what was being done with the fractions above.
If we are told that an equation such as
is true, the rules of algebra allow us to manipulate its appearance without changing the “truth” that it represents. As long as we do exactly the same thing to both sides of the equation, the truth of the equation is not broken, and the equation will still represent the same quantitative relationship. So, multiplying both sides of the above equation by 3 produces
This equation sure looks different than the first one, but the same value of x makes them both true. I could add 15 to both sides, and once again the resulting equation looks quite different, but is made true by the same value of x:
All three of the above equations are equivalent to each other. And the same key concepts I mentioned for equivalent fractions apply:
a) an infinite number of equivalent equations can easily be created, and
b) while all these equivalent equations sure look different, the solution to any one of them is the solution to all of them
This should reassure those who get nervous about how to start an algebra problem, as I will explain.
Many students seem to have the impression that there is only one way to do a math problem, and get nervous when they don’t recall the details of how they are “supposed to” start a problem. The concept of “equivalent equations” as described above provides a way to get around this stumbling block: it does not matter what step you take first, as long as you follow the rules of algebra. Even if you happen to choose a very “in-efficient” path to an answer, if you follow the rules of algebra you will still get a correct answer.
As a example, suppose you are asked to solve:
Your first step could be to add 6 to both sides, perhaps “because you like 21 more than 15 as a number”:
Did this first step take me closer to an answer? No – this equation doesn’t really look any closer to a solution than the original one. But notice that all values of “x” that make this equation true will also make the previous equation true, so adding 6 to both sides was NOT “wrong”… it produced an equivalent equation. At this point, you could add (2x) to both sides:
This looks even worse. Why does it look worse? Because there are now more terms than there were originally, and we were trying to reduce the number of terms in order to get x all by itself on one side. With that thought in mind, why not move the 21 to the other side by subtracting 21 from both sides:
Then subtract 2x from both sides:
And now divide both sides by 3:
Even though the above example was NOT solved in an efficient way, it still arrived at the correct solution via a series of equivalent equations. So, be careful that you are following the rules of algebra at each step (so that each step is equivalent to the previous one), but don’t worry about memorizing the steps needed to solve a particular type of problem. You will find it more useful to think about how you want the equation to look when you are finished, then make a series of small changes – one at a time – that seek to get it looking more and more like the “final form” you have in mind.
The more problems you solve using algebra, the more confident and efficient your work will become. Your teachers will usually seek to teach you the most efficient path to a solution, but if you don’t happen to remember it – don’t worry about it. Just as you don’t have to take the most direct route when travelling somewhere (many prefer to take a longer scenic route), you don’t have to solve a math problem in the most efficient way. In fact, much real world problem solving is often not very efficient. So don’t worry about efficiency when you are learning something new – it will come with experience. For now, focus on successfully and confidently getting to a correct solution. If you happen to take a longer path to get there than a classmate, perhaps you will have seen more interesting sights (learned a bit more) than they did along the way.
So two equations are equivalent when they have the same solutions, right? This is a nice way to characterize the symbolic work of algebra. It’s mathematically correct, and it’s intellectually honest. As you noticed in my blog entry titled CAUTION! (link below), I believe both of these to be important criteria.
And using “equivalence” in more contexts than just fractions is surely a good thing for the betterment of mathematics learning.
I do feel that your definition of algebra is limited. I appreciate its spirit-it is offered to students as an attempt to demystify this abstract subject-but is algebra really “a set of rules”? Or is it more a set of habits of mind, such as (1) looking for ways to generalize particular computations, or (2) wondering whether particular results are always true? We cannot do these things without some rules, to be sure, but I’m not convinced that algebra IS these rules.
Just to ensure nobody mis-interprets your first sentence: equations that share some (but not all) solutions are NOT equivalent… but if the set of all solutions to one equation is the same as the set of all solutions of the others, then they must be equivalent equations.
I think the concept of “an algebra” is a challenging one for students to grasp – particularly those students who are just beginning to move away from a rote memorization approach in favor of conceptual linkages between topics.
From a student’s perspective, I think it is reasonable to describe the properties of addition, subtraction, etc. as a set of rules… until such time as we can turn their attention to questioning the rules and their origins: are they axioms or theorems, what domains do they apply to, etc.