Algebra 1 Equation Manipulations
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Peel the Onion
Equation Manipulations

 

 Let's say we have a problem like:

 

3X - 5 = 4

 
 We need to get this to be ...
 

X = SOME NUMBER

 
 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 ...
 

 

 
 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.
 

 
 Let's try a nasty one ...
 
 Example:
 

 
 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 ...
 

 
 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:
 

 
 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 ...
 

 
 Just with these two rules, 
 we can deal with some really messy stuff.
 
 Let's try this one.
 
 Example:
 

 
 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.
 

 
 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.
 

 
 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 ...
 

 
 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.

 To get rid of it, we multiply both sides by 4/1 ...

 

 
 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 ...

 

 
 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 ...
 

 
 We can do a bunch of stuff with those two little rules!
 

   copyright 2005 Bruce Kirkpatrick

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