Question 134886: A company makes three types of backpacks for a cost of $754 for 100 backpacks. Production costs for the three types are $5, $8, and $12. They sell for $20, $30, and $50 respectively. How many of each type are made if the profit is $2226?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A company makes three types of backpacks for a cost of $754 for 100 backpacks. Production costs for the three types are $5, $8, and $12. They sell for $20, $30, and $50 respectively. How many of each type are made if the profit is $2226?
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x,y,z = no. of each type of backpack
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Total backpacks equation
x + y + z = 100
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Total cost equation
5x + 8y + 12z = 754
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We want the last equation to be the total revenue, add profit to cost:
754 + 2226 = 298
So we have:
20x + 30y + 50z = 2980
Simplify, divide eq by 10
2x + 3y + 5z = 298
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Use elimination to get two 2 unknown equations:
Multiply 1st equation by 2 and subtract from the last equation
2x + 3y + 5z = 298
2x + 2y + 2z = 200
---------------------subtracting eliminates x
0x + 1y + 3z = 98
y + 3z = 98; our 1st two unknown equation
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Multiply the 1st equation by 5 and subtract it from the 2nd equation:
5x + 8y + 12z =754
5x + 5y + 5z = 500
---------------------subtracting eliminates x
0x + 3y + 7x = 254
3y + 7x = 254; our 2nd two unknown equation
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Multiply the 1st two unknown equation by 3, subtract the 2nd two unknown equation:
3y + 9z = 294
3y + 7z = 254
--------------subtracting eliminates y
0y + 2z = 40
z = 40/2
z = 20 ea $50 backpacks
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Find y using the 1st two unknown equation
y + 3(20) = 98
y = 98 - 60
y = 38 ea $30 backpacks
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Find x using the 1st equation:
x + 38 + 20 = 100
x = 100 - 58
x = 42 ea $20 backpacks
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Check solution using the total cost equation
5(42) + 8(38) + 12(20) =
210 + 304 + 240 = 754; confirms our solution
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A lot of steps but all of them pretty simple. Hope it made sense to you.
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