| In the last few pages, we have been working with things called equations. | ||
| Oh yeah? What's an equation? | ||
| AN EQUATION IS SOMETHING WHERE YOU HAVE | ||
| AN EQUAL SIGN BETWEEN TWO GROUPS OF STUFF | ||
| That is ... | ||
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STUFF = STUFF |
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| Like: | ||
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4X + 5 = 12 |
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| If we just have some terms and no equal sign | ||
| THAT is called an EXPRESSION. | ||
| Expression - equation - what's the difference? | ||
| PLENTY! | ||
| When we have an EQUATION | ||
| we can do almost anything we want to it to solve it. | ||
| We just have to do the same thing to both sides of the equation. | ||
| When we have an expression, we don't have two sides. | ||
| We just have one thing. | ||
| We can't do anything that would change the value of the expression. | ||
| That means all we can do is multiply it by 1 or add 0. | ||
| That doesn't sound like very much. | ||
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No it isn't. But using some tricks, it's often enough. |
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| Old Time Review: | ||
| What would you do if you had: | ||
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| We would find a common denominator and subtract. | ||
| There are lots of ways to find a common denominator. | ||
| The easiest way is to just call it the product | ||
| of the two denominators. | ||
| The denominators are 3 and 7, so the common denominator | ||
| is 3 x 7 = 21. | ||
| Since this is an expression, not an equation | ||
| we can only multiply it by one or add zero. | ||
| Here we will multiply the first term by 7/7 | ||
| and multiply the second term by 3/3 ... | ||
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| That gives us our common denominator, | ||
| and we can subtract. | ||
| NOW BACK TO ALGEBRA | ||
| Say we have this ... | ||
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| We need to do the same process. | ||
| The easiest common denominator is (X - 3)(X + 3). | ||
| Multiply each term by the name for 1 | ||
| that gets us a common denominator | ||
| subtract and simplify ... | ||
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| Did you catch the signs on the subtraction? | ||
| Now simplify ... | ||
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| If we had a problem that used addition instead of subtraction | ||
| it would work the same way ... | ||
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| OK. Let's try another one ... | ||
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| so at first, this one is a problem. | ||
| The term on the right is just an X. | ||
| The deal is, we can put a denominator of 1 | ||
| under anything so ... | ||
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| so: | ||
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| The common denominator is 4 × 1 = 4. | ||
| The term on the left already has this denominator, | ||
| so we just need to multiply the term on the right by 4/4. | ||
| Then we can add and simplify ... | ||
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| There's nothing special about X. | ||
| It's just the variable we tend to use. | ||
| Try this ... | ||
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| The first step is to find a common denominator. | ||
| The first term denominator is Ho. | ||
| The second term denominator is Hi. | ||
| That makes the common denominator Hi x Ho. | ||
| Math people might write this as Hi Ho. | ||
| The first term denominator needs Hi. | ||
| The second term denominator needs Ho. | ||
| We don't have an equation, | ||
| so we can only multiply terms by another name for 1. | ||
| The first term gets multiplied by Hi/Hi. | ||
| The second term gets multiplied by Ho/Ho. | ||
| We get ... | ||
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| Now it's time for a math joke ... | ||
| Can you factor a worm out of an apple? | ||
| Sure you can. | ||
| You want to start with | ||
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Apple |
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And end up with ... |
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Apple = Worm × SOMETHING |
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| Whatever that something is, | ||
| when you multiply it times Worm you get apple. | ||
| Can you figure out what it is??? | ||
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copyright 2005 Bruce Kirkpatrick |
