document.write( "Question 182562: given: stamps of 0.5, 3, 10 dollar
\n" ); document.write( "Question: how many stamps from each do i need to have 100 stamps for a total of 100$ \n" ); document.write( "
Algebra.Com's Answer #137042 by ptaylor(2198)\"\" \"About 
You can put this solution on YOUR website!
WE KNOW RIGHT OFF THE BAT THAT WE HAVE THREE UNKNOWNS AND ONLY TWO EQUATIONS SO MORE THAN ONE ANSWER IS A DISTINCT POSSIBILITY
\n" ); document.write( "Let x=number of 0.5 dollar stamps
\n" ); document.write( "y=number of 3 dollar stamps
\n" ); document.write( "z=number of 10 dollar stamps\r
\n" ); document.write( "\n" ); document.write( "x+y+z=100-----------------------eq1
\n" ); document.write( "0.5x+3y+10z=100----------------eq2\r
\n" ); document.write( "\n" ); document.write( "multiply eq2 by 2 and subtract eq 1 from it:
\n" ); document.write( "(x+6y+20z=200)-(x+y+z=100) and we get
\n" ); document.write( "5y+19z=100 solve for y
\n" ); document.write( "5y=100-19z
\n" ); document.write( "y=(100-19z)/5 or
\n" ); document.write( "y=20-19z/5----------------------eq2a
\n" ); document.write( "Now we know two important aspects of this problem:
\n" ); document.write( "(1) We cannot have fractions of dollars other than the half dollars
\n" ); document.write( "(2) We cannot have negative dollars
\n" ); document.write( "So, from eq2a, 20-19z/5 >=0
\n" ); document.write( "-19z/5>=-20
\n" ); document.write( "-19z>=-100
\n" ); document.write( "19z<=100
\n" ); document.write( "z<=5.263----round to the next smallest whole number
\n" ); document.write( "z<=5
\n" ); document.write( "Now we will start assigning values to z between 0 & 5 and see which values produce whole numbers for y in eq2a.
\n" ); document.write( "z=0 y=20--------------------possibility
\n" ); document.write( "z=1 y=fraction--------------------no good
\n" ); document.write( "z=2 y=fraction ---------------------no good
\n" ); document.write( "z=3 y=fraction-----------------------no good
\n" ); document.write( "z=4 y=fraction------------------------no good
\n" ); document.write( "z=5 y=1------------------------possibility\r
\n" ); document.write( "\n" ); document.write( "Only two possibilities:
\n" ); document.write( "z=0, y=20 or z=5, y=1
\n" ); document.write( "Now we plug these possibilities in eq1 and see what we get for x
\n" ); document.write( "y=0, z=20
\n" ); document.write( "x+20+0=100
\n" ); document.write( "x=80
\n" ); document.write( "So now we have x=80, y=20 & z=0
\n" ); document.write( "Let's see if these work in eq 2:
\n" ); document.write( "0.5*80+3*20+10*0=100
\n" ); document.write( "40+60=100
\n" ); document.write( "100=100
\n" ); document.write( "So, one solution is:
\n" ); document.write( "x=80, y=20 & z=0----------------------------------One solution
\n" ); document.write( "Now for y=1, z=5
\n" ); document.write( "Plug these into eq1 and we get:
\n" ); document.write( "x+1+5=100
\n" ); document.write( "x=94
\n" ); document.write( "Plugging these into eq2:
\n" ); document.write( "0.5*94+3*1+5*10=100
\n" ); document.write( "47+3+50=100
\n" ); document.write( "100=100
\n" ); document.write( "So x=94, y=1, z=5-------------------------------Another solution\r
\n" ); document.write( "\n" ); document.write( "Hope this helps-------ptaylor\r
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