



It
would be great if every time we had to solve a problem,



somebody
just gave us the equation to use. 


Well
guess what, 


that
doesn't always happen. 


Lots of
the time we get a little story that 


we have
to translate into an equation. 





This
whole thing usually starts with two values that change. 


Usually
one of the things can change on its own 


and the
other thing changes BECAUSE the first thing changed. 


That is,
if the first thing hadn't changed, 


the
second thing wouldn't have changed. 





The thing
that can change by itself 


is called
the independent variable. 


It does
what it wants, it's INDEPENDENT. 


The thing
that changes BECAUSE the first thing changed 


is called
the dependent variable. 


What it
does depends on what the first thing does, 


it is
DEPENDENT on the first thing. 





In math,
the thing that can do what it wants is often called X. 


and the
thing that changes depending on what X does is often called Y. 





Sometimes
when X gets bigger so does Y. 





If when X
gets bigger Y gets bigger too, 


it is
called direct variation. 


In math
talk we say "Y varies directly as X." 


In
English we say "If X goes up, so does Y. If X goes down, so
does Y." 





If Y
varies directly as X and they both have the same value 


the
equation for this is: 


Y
= X 





Even if X
and Y are both going up or down, 


They
usually do not have the same value . 





Example: 


If I am
traveling down the road at 50 miles per hour, 


after 1
hour, time has changed by 1 hour. 


but miles
traveled has changed by 50 miles. 





We can
change our Y = X (varies directly) equation 


to show
this relation. 


As X
changes by 1 Y changes by 50, 


so we
just need to multiply X. 





We can
fix our Y = X equation to do this. 


We make
it: 


Y = 50X






In
general, we call this form: 


Y
= kX



Where k
stands for any number. 





Example: 


Y varies
directly as X. 


When Y =
2, X = 10. 


Find k
... 








So for
this problem, the equation is ... 








Example: 


For the
problem in example 1, find Y when X = 25. 








One thing
you can probably see is that in this problem, 


when X =
0 then Y = 0 too. 


Sometimes
"Y varies directly as X", But when X = 0, Y = not zero. 


This is
almost the same thing, but leave it to math people 


to give
it it's own special name. 


They say: 


"Y
varies linearly with X" 


In
general, the equation looks like: 





Y = kX + C






We have 4
things here, X, Y, k, and C. 


To solve
this one, we need to know a lot ... 





Example: 


Y varies
linearly with X. When X = 0, Y = 5. When X = 3, Y = 11. 


Find the
entire equation. 





OK,
taking one piece at a time ... 





Y
= kX + C






When X =
0, Y = 5 so ... 


Y
= kX + C



5
= k0 + C



5
= C






So now we
have ... 


Y
= kX + 5






We still
need to find k. 


We are
told that when X = 3, Y = 11. 


So put
those numbers in and see what you get ... 











So we
have: 


Y
= 2X + 5






Example: 


Y varies
linearly with X. 


When X =
2, Y = 15. 


When X =
4, Y = 23 





OK, here
we don't know the value of Y when X = 0 


so we
need more tricks ... 


What we
do is put in the first X and Y values and get one equation, 


Then put
in the second X and Y values and get a second equation. 





Y
= kX + C 
Y
= kX + C 
Y
= 15, X = 2 
Y
= 23, X = 4 
15
= k(2) + C 
23
= k(4) + C 
15
= 2k + C 
23
= 4k + C 






What we
have now is 2 equations with 2 unknowns... 


The only
weird thing is that the unknowns are k and C 


rather
than X and Y. 


But k and
C and X and Y are just names. 


It really
doesn't make any difference to the math 


if you
use an X or a Q or a J or whatever ... 





OK, so 2
equations with the same two unknowns. 


We can
rewrite both equations as C = stuff 


and then
substitute the stuff in 1 equation 


for the C
in the other. 


Then we
will have an equation with 1 unknown. 


That we
can solve! 


Here goes
... 








Now find
C ... 


15
= 2k + C



15
= 2(4) + C



15
= 8 + C



7
= C






So C = 7
and k = 4. 


That
means the equation is ... 





Y
= kX + C



Y
= 4X + C






copyright 2005 Bruce Kirkpatrick 
