



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 
