



If
we have two equation lines with different slopes, 


the
cross ... SOMEWHERE. 





For
example, the equations: 





Y = 3X 
5 and Y = X + 1 





Solving
these for X = 0, and X = 2 we get ... 





(X
= 0) 
Y = 3(0)  5 

Y = (0) + 1 

Y = 5 

Y = 1 




X
= 0, Y = 5 
X
= 0, Y = 1 




(X
= 2) 
Y = 3(2)  5 

Y = (2) + 1 

Y = 1 

Y = 3 




X
= 2, Y = 1 
X
= 2, Y = 3 






For
Y = 3X  5, two points on the equation line 


are
X = 0, Y = 5 and X = 2, Y = 1. 





For
Y = X + 1, two points on the equation line 


are
X = 0, Y = 1 and X = 2, Y = 3. 





So
we can draw the graphs of these two equations ... 











We
can see from the picture that the two lines do cross. 


If
we draw the picture perfectly 


and
really, really big, 


we
can maybe tell exactly where the lines cross. 





But
there's a better way ... 





At
the point where the lines cross 


the
X and Y values of the lines are the same numbers. 





Well so
what?



What
good does that do? 





Watch
closely ... 





If
the Y values are the same for both equations, 


we
can link the two equations on the same Y. 





Y = 3X 
5 and Y = X + 1 


3X  5 = Y
= X + 1 





Actually,
we don't even need the Y right now, 


so
we can get rid of it ... 





3X  5 = X + 1 





We've
solved stuff like this before ... 





STEP
1 


Get
all of the X's on one side 





Here
that means subtracting an X from each side ... 











STEP
2: 


Get
all of the numbers on the other side 





Here,
that means adding 5 to each side ... 











STEP
3 


Peel
the number (the coefficient) away from the X 





Here
that means dividing both sides by 2. 











The
point where the two lines cross 


is
the point where X = 3. 





Now
that we have the value of X, 


we
can go back to either equation 


and
get the value for Y. 





Or
do them both for fun. 


FOR
FUN???? 





(X
= 3) 



Y = 3X  5 
Y = X + 1 

Y = 3(3)  5 
Y = 3 + 1 

Y = 9  5 
Y = 4 

Y = 4 







So
the place where the two equation lines cross 


is
exactly the point: X = 3, Y = 4. 





We
sometimes write that point (3,4) 











This
point is called the solution "set" of the equations. 





That
means it is the one and only point 


that
is part of both of the lines. 





Which
means it is the point where the two lines cross. 





When
we did the calculation, 


we
joined the two equations on the Y. 


We
COULD have joined them on the X instead. 


Do
it whichever way is easier for you. 


You
get the same answer either way. 





copyright 2005 Bruce Kirkpatrick 
