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Let's say we have a problem like: |
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3X - 5 = 4 |
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| We need to get this to be ... | ||
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X = SOME NUMBER |
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| When we get the problem, | ||
| there will be stuff all around the X. | ||
| Basically, solving the problem means | ||
| peeling all the stuff away from the X. | ||
| It's kind of like peeling the layers in an onion. | ||
| Here's how we do it. | ||
| Start with the thing that's the farthest away from the X | ||
| on the same side of the equation as the X | ||
| and then work your way in towards the X. | ||
| OK, I give up. What's "farthest away from the X" mean? | ||
| Any term with no X in it on the same side as the X | ||
| is the farthest away. | ||
| Remember that terms are groups of stuff | ||
| that are completely separated from each other | ||
| by a plus or a minus sign. | ||
| That means in the problem we got at the top of the page: 3X - 5 = 4 | ||
| the farthest away term is that -5. | ||
| To peel the -5 away from the X, | ||
| add 5 to both sides ... | ||
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| So now we have 3X = 9. | ||
| We've seen stuff like that before. | ||
| We need to peel the 3 away from the X. | ||
| The 3 is multiplied times the X | ||
| so to get rid of it, we need to divide both sides by 3. | ||
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| Let's try a nasty one ... | ||
| Example: | ||
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| OK, any term with no X in it, | ||
| on the same side as the X, is the farthest away. | ||
| So let's "peel" away that -2 ... | ||
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| NOW WHAT? | ||
| Now we need a new rule. | ||
| Relax, we only have 2 rules total. | ||
| Here's rule number 2: | ||
| When there is only 1 term on the side with the X | ||
| anything on the other side of the biggest fraction line | ||
| is farthest away from the X. | ||
| That means we peel away the 5 in the denominator next. | ||
| If we just think about that 5 in the denominator, | ||
| it is really like 1/5. | ||
| So to turn it into a 1, we multiply by 5/1. | ||
| and anything we do to one side, | ||
| we have to do to the other side. | ||
| So: | ||
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| Now we're getting there. | ||
| We just need to peel away that 4. | ||
| The 4 is multiplied times the X, | ||
| so to get rid of it we divide by 4. | ||
| And anything we do to one side | ||
| we need to do to the other side ... | ||
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| Just with these two rules, | ||
| we can deal with some really messy stuff. | ||
| Let's try this one. | ||
| Example: | ||
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| First, let's review our two rules: | ||
| RULE 1: | ||
| A term with no X on the side with the X | ||
| is farthest away from the X. | ||
| RULE 2: | ||
| When there's only one term, | ||
| the stuff on the other side of the biggest fraction line | ||
| from the X is the farthest away. | ||
| We have two terms on the side with the X, | ||
| so we know it's not time for rule 2 yet. | ||
| The -4 is our first target. | ||
| Since it is a minus, | ||
| we "peel" it away by adding 4 to both sides. | ||
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| Now that there's just one term on the left side of the equation | ||
| it is time for rule 2. | ||
| The 6 is the thing on the far side of the biggest fraction line, | ||
| so it gets peeled next. | ||
| A 6 in the denominator is really a 1/6, | ||
| so we get rid of it by multiplying by 6/1. | ||
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| So now we have peeled a layer off of the onion | ||
| and are back to two terms on the left side of the equation. | ||
| The "- 1" term has no X in it | ||
| so it gets peeled away ... | ||
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| We only have one term on the left side | ||
| of the equation now. | ||
| The 4 is on the far side of the biggest (and only) fraction line | ||
| so it gets peeled away next. | ||
| Since the 4 is in the denominator it is really a 1/4. | ||
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To get rid of it, we multiply both sides by 4/1 ... |
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| So we've peeled things away | ||
| to where we're back to two terms on the left side. | ||
| The "+ 7" is a term with no X in it, | ||
| so it goes away next ... | ||
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| All that's left on the side with the X is the 3. | ||
| The three is multiplied times the X | ||
| so to peel it away, we divide both sides of the equation by 3 ... | ||
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| We can do a bunch of stuff with those two little rules! | ||
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copyright 2005 Bruce Kirkpatrick |
