If you are having difficulty with algebra, these pages may be of some help because they offer different sorts of explanations, in some sense more "psychologically complete", than are usually found in algebra texts.
It is my belief that the way algebra is typically presented to students leaves out some ideas and explanations that are helpful, even sometimes necessary, for them to be able to do algebra well and to have a "feel" for it.
There are at least three different kinds of things taught in algebra courses: (1) language conventions, (2) logical numerical manipulations using those conventions, and (3) deducing answers to problems by using the conventions and the logical manipulations of them. I will explain as I go. But it is important for students to keep in mind whether in a given lesson they are supposed to be learning a convention, a manipulation, or a way of solving problems by using conventions and manipulations.
It is also my belief that school "culture" is such that, even contrary to good teachers' warnings, students will often think they are simply supposed to memorize formulas and recipes in math, rather than (also) understand them. Such memorization becomes a problem in courses, such as algebra, where understanding is at least as important as specific knowledge.
This is a two-fold problem. (1) Teachers need to try to give useful and helpful explanations -- and they need to be aware of as many typical student misunderstandings and confusions as they can; and teachers need to constantly try to monitor students for confusion and misunderstandings about what has been presented; waiting until there are test results is often too late. But also, (2) students need to know that THEY are the ones who will have to ultimately make the material make sense to them, and that they need to keep trying until it does. They may have to consult others, find a different book, or just sit down and think about the material, if they cannot understand their teacher's explanation about some aspect of algebra or other. There is simply no guaranty that any particular explanation will provide automatic understanding. Understanding requires reflective thinking of one's own. Explanations often are only a help to such thinking; and what serves as a great explanation for one student may not be helpful at all to another.
(My own first difficulty in "pre"-algebra was not understanding what letters such as "x" had to do with anything, and why letters were chosen to represent quantities at all, or how you worked with them when you had them. I vividly remember when the light dawned on me about this particular lack of understanding, in part because I still do not know why or how the teacher's particular explanation "worked" on me. She was saying that doing algebra was like unwrapping a package in the reverse way it was wrapped to begin with. It may be that I figured out what I needed to know while she was talking instead of because of what she specifically said, or it may be that what she said had some sort of meaning to me subconsciously somehow. I don't know, since "unwrapping" is not the way I see (or even then saw) algebra. But what follows are explanations of the sort that seem to me the most meaningful about some aspects of algebra many students typically have trouble with. Further explanations of other aspects of algebra can be found at A supplemental introduction to the first chapter of an algebra book and at Rate, Time Problems.)
The following question was asked on the Math-Help forum. It is typical of the kinds of problems had by students who don't really understand in general "what is going on" in algebra -- why you do certain manipulations of formulas, or how you choose which manipulations to do. Particular algebraic manipulations do not make sense to them because they don't have a general sense of what algebra is about, or what the point of the manipulations is. After I give the response to this question, a response which will include both general and specific problem-solving ideas, I will make come comments about how a typical algebra course is structured, and why it is structured that way.
I have a big exam Monday in algebra and I have
no idea how to do linear equations! Can someone please help quick? Her
are examples of a few that I am having problems with.
3(3 - 4x) + 30=5x - 2(6x-7) and 5x²-[2(2x²+3)]-3=x²-9
I am also having a few problems with this:
x + 3 + 2x = 5 + x + 8 (5)
I am supposed to figure out if 5 is the answer, and tell how I got the answer.
It looks to me from this last problem that you perhaps don't have an UNDERSTANDING of what doing algebra with equations is all about, which makes doing any problems a bit difficult. But let's see what we can do for you here. The following may be too much for you to absorb before your test, though I hope not; but I think it is stuff you will need to know for future tests as well, so maybe it will help you for them even if it is too much for tomorrow's test.
Take the last problem first. Do you understand that if "five is the answer," that simply means that IF YOU WERE TO SUBSTITUTE 5 AS THE VALUE EVERYWHERE THERE IS AN X, THE STATEMENT THAT THE LEFT SIDE OF THE EQUATION EQUALS THE RIGHT SIDE WOULD BE TRUE? If 5 is NOT the solution, then when you make the substitution, the statement will not be true that the left side of the equation is equal to the right side.
Take a simple case first: X = 27 - 4
There is only one number X can be for this statement to be true; 23, right? So 19 would NOT be a solution to this equation; that is, X cannot equal 19 and the statement still be true that the left side equals the right side.
Now make it just a bit harder: X + 3 = 27 -
There is still only one number this can work for but it is a different number, because now we know that it is not X that equals 23, but some number that, when you add 3 to it, gives you 23. So what number gives you 23 when you add 3 to it? 20, right? That means X must be 20 in that equation.
It gets a little harder when you start putting X's on both sides of the equations and add or subtract some multiplications and divisions, etc., but the idea of what is going on is the same.
So if we look at the equation you gave last: if X does equal 5 in the equation x + 3 + 2x = 5 + x + 8, then that would mean