document.write( "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? \n" ); document.write( "
Algebra.Com's Answer #98785 by ankor@dixie-net.com(22740)\"\" \"About 
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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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\n" ); document.write( "x,y,z = no. of each type of backpack
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\n" ); document.write( "Total backpacks equation
\n" ); document.write( "x + y + z = 100
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\n" ); document.write( "Total cost equation
\n" ); document.write( "5x + 8y + 12z = 754
\n" ); document.write( ":
\n" ); document.write( "We want the last equation to be the total revenue, add profit to cost:
\n" ); document.write( "754 + 2226 = 298
\n" ); document.write( "So we have:
\n" ); document.write( "20x + 30y + 50z = 2980
\n" ); document.write( "Simplify, divide eq by 10
\n" ); document.write( "2x + 3y + 5z = 298
\n" ); document.write( ":
\n" ); document.write( "Use elimination to get two 2 unknown equations:
\n" ); document.write( "Multiply 1st equation by 2 and subtract from the last equation
\n" ); document.write( "2x + 3y + 5z = 298
\n" ); document.write( "2x + 2y + 2z = 200
\n" ); document.write( "---------------------subtracting eliminates x
\n" ); document.write( "0x + 1y + 3z = 98
\n" ); document.write( "y + 3z = 98; our 1st two unknown equation
\n" ); document.write( ":
\n" ); document.write( "Multiply the 1st equation by 5 and subtract it from the 2nd equation:
\n" ); document.write( "5x + 8y + 12z =754
\n" ); document.write( "5x + 5y + 5z = 500
\n" ); document.write( "---------------------subtracting eliminates x
\n" ); document.write( "0x + 3y + 7x = 254
\n" ); document.write( "3y + 7x = 254; our 2nd two unknown equation
\n" ); document.write( ":
\n" ); document.write( "Multiply the 1st two unknown equation by 3, subtract the 2nd two unknown equation:
\n" ); document.write( "3y + 9z = 294
\n" ); document.write( "3y + 7z = 254
\n" ); document.write( "--------------subtracting eliminates y
\n" ); document.write( "0y + 2z = 40
\n" ); document.write( "z = 40/2
\n" ); document.write( "z = 20 ea $50 backpacks
\n" ); document.write( ":
\n" ); document.write( "Find y using the 1st two unknown equation
\n" ); document.write( "y + 3(20) = 98
\n" ); document.write( "y = 98 - 60
\n" ); document.write( "y = 38 ea $30 backpacks
\n" ); document.write( ":
\n" ); document.write( "Find x using the 1st equation:
\n" ); document.write( "x + 38 + 20 = 100
\n" ); document.write( "x = 100 - 58
\n" ); document.write( "x = 42 ea $20 backpacks
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\n" ); document.write( "Check solution using the total cost equation
\n" ); document.write( "5(42) + 8(38) + 12(20) =
\n" ); document.write( "210 + 304 + 240 = 754; confirms our solution
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\n" ); document.write( "A lot of steps but all of them pretty simple. Hope it made sense to you.
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