



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
... 


STUFF = STUFF 





Like: 


4X + 5 = 12 





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. 





No it
isn't. But using some tricks, it's often enough.






Old
Time Review: 





What
would you do if you had: 











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 ... 











That
gives us our common denominator, 


and we
can subtract. 





NOW BACK
TO ALGEBRA 





Say we
have this ... 











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 ... 











Did you
catch the signs on the subtraction? 





Now
simplify ... 








If we had
a problem that used addition instead of subtraction 


it would
work the same way ... 











OK. Let's
try another one ... 











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 ... 





so: 








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 ... 











There's
nothing special about X. 


It's just
the variable we tend to use. 





Try this
... 











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
... 











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 


Apple 





And end
up with ...



Apple
= Worm × SOMETHING






Whatever
that something is, 


when you
multiply it times Worm you get apple. 


Can you
figure out what it is??? 











copyright 2005 Bruce Kirkpatrick 
