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Question 61637: The local yogurt bar features a banana treat made up of 2lb of bananas, 3lb of topping , and 4lb of frozen yogurt. The cost of the banana treat is $19.00. One pound of topping costs $1 less than one pound of frozen yogurt, which costs as much as 1/2 pound of topping and 4 pounds of bananas. How much does one pound of each ingredient cost?
Answer by ptaylor(2198)   (Show Source):
You can put this solution on YOUR website! Let x=cost of bananas ($)
Let y=cost of topping ($)
Let z=cost of frozen yogurt($)
Now we know that the cost of 2 lb of bananas (2x) plus the cost of 3 lb of topping (3y) plus the cost of 4 lb of frozen yogurt (4z)equals $19.00. This gives us one of the three equations that we need in order to solve this problem.
(1) 2x+3y+4z=19
We are told that one lb of topping (y) costs $1 less that one lb of frozen yogurt (z). This gives us our second equation:
y=z-1 or
(2) y-z=-1
We are also told that the cost of frozen yogurt (z) is the same as the cost of 1/2 lb of topping (1/2y) plus 4 lb of bananas (4x). Now we have our third equation:
z=(1/2)y+4x or
(3) 8x+y-2z=0
Putting our three equations together, we have:
(1) 2x+3y+4z=19
(2) y-z=-1
(3) 8x+y-2z=0
I will solve these by simply manipulating the equations and you can solve them by the use of matrices.
First, solve for z in (2) and substitute into (1) and (3) and we now have:
(1a) 2x+3y+4(y+1)=19 and 2x+7y=15
(3a) 8x+y-2(y+1)=0 and 8x-y=2
Solve for y in (3a) and substitute into (1a) and we get:
(1b) 2x+7(8x-2)=15 simplifying we get:
58x=29
x=$.50 cost of bananas
From (3a) 8x-y=2 or (8)(.50)-y=2
y=$2.00 cost of topping
From (2) above we have y-z=-1 or z=y+1
Substuting $2.00 for y, we have:
z=$3.00 cost of frozen yogurt
Hope this helps-----ptaylor
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