Q&A #7470

Teachers' Lounge Discussion: System of equations

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From: Tara Sorace

To: Teacher2Teacher Public Discussion
Date: 2002121701:49:38
Subject: discussion

A paper manufacturer has two factories. One factory produces 800 reams of medium grade paper and 300 reams of high grade paper per day. The second 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 both 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 to 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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