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Sumatra costs $2.50 per with colombian coffee which costs $3.75 pound. wants to make 50 pounds cost $3.35 per pound. how many pounds of each will he need?

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Tamara J. | Math Tutoring - Algebra and Calculus (all levels)Math Tutoring - Algebra and Calculus (al...
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Let 'x' represet the number of pounds of sumatra, which costs $2.50 per pound, and 'y' represent the number of pounds of columbian coffee, which costs $3.75 per pound. The problem asks you to find how many pounds of each kind of coffee you will need to combine to make 50 pounds of coffee that will cost $3.35 per pound.  

With this, we know the following information:

     Sumatra coffee = $2.50/lb

     # of lbs of Sumatra coffee = x

     Columbian coffee = $3.75/lb

     # of lbs of Columbian coffee = y

     Cost of combo of coffees = $3.35/lb

     # of lbs of coffees combined = 50 lbs

This problem calls for a system of linear equations. One of the equations can be generated by combining the number of pounds of each kind of coffee needed to equal to total number of pounds we need to make the coffee combo. The other equation combines the cost of each kind of coffee, found by multiplying the cost of each kind by the number of pounds needed, to equal the total cost of the coffee combo for the total number of pounds given. That is,

     # of pounds of sumatra + # of pounds of columbian = # of pounds of coffee combo

            x  +  y  =  50

(cost of sumatra·# of lbs of sumatra)+(cost of columbian·#of lbs of columbian)=(cost of combo·# of lbs of combo)

    (2.50·x)  +  (3.75·y)  =  (3.35·50)

So, our system of equations consists of the following equations:

             x   +   y   =  50

     (2.50)x  +  (3.75)y = 167.50

We solve for the system by solving for one of the variables first, then using the value for the now known variable to solve for the other variable. One method to solve for the system is the elimination method, by which we eliminate one variable by manipulating one or both equations, if needed, then combining the two to solve for the remaining variable. Once we've solved for one variable we plug its value back into one of the original equations to solve for the other variable.

For simplicity, let's first eliminate x and solve for y by multiplying the entire first equation by -2.50 then adding it to the second equation:

     -2.50·( x + y = 50)     ==>     (-2.50)x + (-2.50)y = (-2.50)(50)

                                                 (-2.50)x - (2.50)y  =  -125

     (-2.50)x  -  (2.50)y  =  -125

+   (2.50)x  +  (3.75)y  = 167.50


        (0)x  +  (1.25)y  =  42.50

                    (1.25)y  =  42.50

     (1.25)y / 1.25  =  42.50 / 1.25

                       y  =  34

Use this value for y and the first original equation to solve for x:

     x + y = 50

     x + 34 = 50

SUbtract 34 rom both sides of the equation to solve for x:

     x = 16

   x = # of lbs of sumatra = 16 lbs

   y = # of lbs of columbian coffee = 34 lbs.

Thus, you will need 16 pounds of sumatra and 34 pounds of columbian coffee to make a 50 pound combo that will cost $3.35 per pound.