Sunday, February 05, 2006

Basic Algebra III

Combining the two principles of algebra to solve more complicated problems

If Ax + B = C, what is x?

Now, we need to be able to use these two principles of algebra together to solve more complicated problems:

If we have numberofbooks books and also a weightofcheese block of cheese in our bag, and it weighs weightofbag, then how much does each book weigh (assuming that the books are all identical)?

So, to solve this, we could break the problem into two steps - what is the weight of all these books together weightofallbooks (= numberofbooks * weightofabook)?

We can determine this:

    weightofallbooks + weightofcheese = weightofbag

    weightofallbooks = weightofbag - weightofcheese

Now that we have the weight of all the books, we can find the weight of a single book:
    numberofbooks * weightofabook = weightofallbooks

    weightofabook = weightofallbooks/numberofbooks

    weightofabook = (weightofbag - weightofcheese)/numberofbooks

Can you see how we've used the two principles I explained earlier to find this result?

Alternately, we can do the whole thing at once. Turning our description into a statement of the weight of the bag:

    numberofbooks * weightofabook + weightofcheese = weightofbag
Then, we simply need to take the required actions so that we are left with "weightofabook = something."

In this case, the required actions are exactly the same as we took above - first you subtract weightofcheese from both sides:

    numberofbooks * weightofabook = weightofbag - weightofcheese.
Then you divide both sides by numberofbooks:
    weightofabook = (weightofbag - weightofcheese)/numberofbooks

The same result as above.

This rule works, in general, for problems of any degree of complication. However, for problems much more complictated that this one, there are no generals, simple, ways to isolate the unknown quantity.

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