Q&A #7470

Teachers' Lounge Discussion: System of equations

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From: Tara Sorace <tara0731@hotmail.com>
To: Teacher2Teacher Public Discussion
Date: 2002121701:49:38
Subject: discussion

A paper manufacturer has two factories. One factory produces 800 reams
medium grade paper and 300 reams of high grade paper per day. The
factory produces 200 reams of medium grade paper and 700 reams of high
grade paper per day. A publishing company has placed an order for 1700
reams of medium grade paper and 2200 reams fo high grade paper. If
factories work together to fill this order, how many days should each
work on the order? Let x = the number of days for the first factory to
work on the order. Let y = the number of days for the second factory
work on the order.
a) Write the two equations
b) Solve the system of equaions
My advice in order to find a process to solve systems of equations
would be to present three different types of methods that are
available.  The problem that you presented in your thread did have a
lot of infomation; however, if you look really carefully, the last two
portions of the information they gave you should really help in
setting up the system. 

You know that for the system of equations that you want to come up
with to equations that have two variables in them (at least in this
problem).  In this case mentioned above, your two variables were x and
y, where x represented the number of days factory #1 has to work to
complete the order and y represents the number of days that factory #2
has to work to complete the order.  So to set this up we need two
equations with some combination of x and y.  

Notice from the given information that the factories are developing
two different types of paper, high and medium grade.  Find out the
amounts of each type of paper that each factory can make a day.  For
example factory 1 produces 800 reams of medium per day and factory 2
produces 200 reams of medium per day.  If factory 1 works x days and
factory 2 works y days, how many days will it take them to fill the
order needed?  Well, in a day, #1 prodcuces 800 reams, so in x days it
will produce 800x reams, likewise in y days, factory #2 will produce
200y reams.  So, if we need to know how long it would take, if both
plants work together to create 1700 reams, then we have the equation
800x + 200y = 1700.  We can use the same process to come up with an
equation for the high grade paper.

As far as solving the system once it has been formed, there are three
different methods that can be used, you can graph both the equations
on the x,y coordinate plane.  Where the two lines meet, they have the
same x and y coordinates, this would be the solution to the system. 
The other two methods are substitution, which involves solving for y
in terms of x or x in terms of y in one equation and then substituting
the value that you solved for into the other equation.  The last
method, which would probably be most useful in the above situation
would be the use of solving by linear combinations.  To do this you
first need to find a least common multiple of either the x or y term. 
Multiply each term by the factor needed so that you can either add or
subtract the equations to eliminate a variable.  After that has
occured, you can solve for the remaining variable and then substitute
back in to find the solution to the system.  Hope this helps.  I
noticed that that problem came from the Focus on Algebra text, there
should be a nice explaination procedure that goes along with that in
the text....just thought I'd let you know!!
Tara Sorace

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