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# A coffee distributor needs to mix a(n) Kenya coffee blend that normally sells for \$11.60 per pound with a they mix?

A coffee distributor needs to mix a(n) Kenya coffee blend that normally sells for \$11.60 per pound with a Queen City coffee blend that normally sells for \$14.30 per pound to create 70 pounds of a coffee that can sell for \$13.91 per pound. How many pounds of each kind of coffee should they mix?

A) Write an equation using the information as it is given above that can be solved to answer the question. Use x as your variable to represent the quantity of Kenya coffee blend.
Equation:

pounds of the Kenya Blend
pounds of the Queen City Blend.

The first part (A) tells you to use x to represent the quantity of Kenya coffee blend, so let's use y to represent the quantity of Queen City coffee blend. Note that these quantities are in pounds.

Since the problem states that these two coffee blends are to be mixed to create 70 pounds of coffee, then we arrive at the following:

quantity of Kenya coffee blend (lbs) + quantity of Queen City coffee blend (lbs)= 70 lbs

OR          x  +  y  =  70

We are given that the Kenya coffee blend sells for \$11.60/lb and the Queen City coffee blend sell for \$14.30/lb. Also, they want to create 70 lbs of coffee that can sell for \$13.91/lb. If we multiply each kind of coffee blend to be mixed by their respective amounts (in pounds) to be mixed then their sum should be equal to the total cost of the new coffee created, which is the selling price for the new coffee multiplied by the total amount (in pounds) created. That is,

(\$11.60/pound)(x pounds) + (\$14.30/pound)(y pounds) = (\$13.91/pound)(70 pounds)

==>     (11.60)x  +  (14.30)y  =  (13.91)(70)

11.6x  +  14.3y  =  973.7

A)        x  +        y  =  70

11.6x  +  14.3y  =  973.7

We now have a system of equations for which we can solve for either by the substitution method or the elimination method. I find the elimination method to be simpler, so I will eliminate one of the variables by multiplying the first equation by a constant that will make one of the variable have a coefficient that is the opposite of the same variable's coefficient in the second equation. For simplicity, I will eliminate x first by multiplying the first equation by -11.6:

-11.6(x + y = 70)     ==>     -11.6x  -  11.6y  =  -812

Now we add this equation to the second equation, eliminating x in the process, then solve for y:

-11.6x  -  11.6y  =  -812

+    11.6x  +  14.3y =  973.7

____________________________

0x  +  2.7y  =  161.7

2.7y  =  161.7

Solve for y by dividing both sides of the equation by 2.7:

2.7y/2.7  =  161.7/2.7

y  =  59.8888

y  ≈  60

Use the value of y found to solve for x using the first original equation:

x  +  y  =  70

x  + 60 =  70

Solve for x by subtracting 60 from both sides of the equation:

x  +  60  -  60  =  70  -  60

x  =  10

Solution:     x = 10     and     y = 60

Thus, they must mix 10 pounds of the Kenya coffee blend and 60 pounds of the Queen City coffee blend.