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- Thread starter Tyrion101
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Mark44

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Do you have an example of this? This is not the way things usually are. More often, what you do in a subsequent class builds on what you've learned in a preceding class. The only thing I can think of that might explain this, is that a way you were taught before is applicable only to certain types of problems, but the new way is applicable to a different set of problems. As you progress in mathematics, theorems play a more pronounced role, and it's important to make sure all of the specified conditions are met before you attempt to use the theorem.For me it's a necessary evil, I need it to do what I want to do, and part if why math usually makes me want to break things is so often when I reach a higher level of math, it often feels as though I'm being taught how to do one thing, like factoring polynomials, and in the next level, I've got to relearn how to do them because I get a wrong answer when I try to do it the way it was done before.

An example would be helpful.

Now I'm sure this is just my perception, but it's part of the reason that I've struggled with math in the past and still struggle at some things. Is this a wrong perception? Is it about how it's taught rather than something else?

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Mark44

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I saw what you posted - why don't you try posting it in the Precalc section under Homework and Coursework?

One problem is that you weren't factoring the complex fractions - you were actually supposed to simplify them, which could mean that you need to factor the numerator or denominator or both. However, whatever you learned before should still be valid, whether you're working with fractions made up of numbers or fractions made up of algebraic expressions. The same ideas apply to both.

A mistake that beginning students make often is cancelling when they shouldn't do so. What cancelling really is, is finding

For example, this is incorrect:

$$ \frac{10 + 8 }{5 + 4} = 2 + 2 = 4$$

This is actually 18/9, which is nowhere near 4.

And this is correct.

$$ \frac{10 + 8}{5 + 4} = \frac{18}{9} = \frac{2 * 9}{9} = \frac{2}{1} * \frac{9}{9}= 2$$

9 is a factor in the numerator and denominator, which means the whole fraction can be thought of as having a factor of 1. Multiplication by 1 always leaves you with exactly the same value.

Things work the same way with variables, except you need to be careful of factors in the denominator that could be zero.

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symbolipoint

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You are in maybe, Introductory Algebra, or Pre-Algebra?

The semester just began recently, since this is September. You become accustomed to the process. You are learning properties of numbers, so as these become more familiar, you will improve at figuring what to do. Most of the time, you look for a way to divide a fraction by 1 or multiply a fraction by 1. Can you give more examples of a problem that gives you trouble?

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So it turns out I was making it more complicated, I was trying to factor things that couldn't be.

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symbolipoint

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So it turns out I was making it more complicated, I was trying to factor things that couldn't be.

Not sure.

Could you provide a couple of examples?

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verty

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So any simplification can be split up into tiny steps that are usually justified by both sides being the same number. Here is another example, we have ##ax = b##, ##cx = d##. We learn that we can subtract one equation from the other, to get ##(a-c) x = b-d##, but why does this work? Well, there is a rule that equals can always be substituted for equals. If two numbers or expressions are equal, one can ALWAYS take the place of the other. So if I start with ##ax - cx = ax - cx##, then on the right, I substitute equals for equals, I get ##(a-c)x = b-d##.

This is my recommendation, find out why each technique works and what steps are involved.

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HallsofIvy

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Mark44

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If you are simplifying an expression, though, you are much more limited in what you can do. You can multiply by 1 (in a variety of forms), or add 0 (also in a variety of forms), or factor the expression, or expand it. In no case can you change the value of the expression.

Looking at the equation that verty showed, ##\frac a b = \frac c d##, one approach is to multiply each side by bd, the common denominator of the two fractions. Another way of looking at this is that you can multiply the fraction on the left by d/d (= 1) and the fraction on the right by b/b (=1). This gives you ## \frac{ad}{bd} = \frac{cb}{bd}##. Since the denominators are equal, and the fractions are equal, it must be that ad = bc.

It sort of goes without saying that neither b nor d can be zero. If either variable were zero, the original equation would be meaningless, and multiplying by b/b or d/d would be meaningless as well.

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Mark44

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What many new students don't realize is that mathematics, unlike normal human conversation, is very terse, and there isn't a lot of redundancy. It's important to pay attention to details - ignoring or missing them will likely cause you to head off in the wrong direction.

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