document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190478>Question 1190478</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a farmer has to buy 100 animals using only $100.<BR>\n" );
document.write( "a chicken costs $0.25<BR>\n" );
document.write( "a sheep costs $3.5<BR>\n" );
document.write( "and a cow costs $10\r<BR>\n" );
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document.write( "how can the farmer buy 100 animals with $100\r<BR>\n" );
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document.write( "Thank you :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #822139 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=822139\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = number of chickens<BR>\n" );
document.write( "y = number of sheep<BR>\n" );
document.write( "z = number of cows<BR>\n" );
document.write( "x,y,z are nonnegative integers\r<BR>\n" );
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document.write( "The farmer must buy 100 animals, so,<BR>\n" );
document.write( "x+y+z = 100\r<BR>\n" );
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document.write( "a chicken costs $0.25<BR>\n" );
document.write( "a sheep costs $3.5<BR>\n" );
document.write( "and a cow costs $10\r<BR>\n" );
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document.write( "Based on those costs, we have<BR>\n" );
document.write( "0.25x = cost of the chickens only<BR>\n" );
document.write( "3.50y = cost of the sheep only<BR>\n" );
document.write( "10z = cost of the cows only\r<BR>\n" );
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document.write( "They sum to<BR>\n" );
document.write( "0.25x+3.50y+10z <BR>\n" );
document.write( "which is the total cost of all the animals.<BR>\n" );
document.write( "This stated total cost is $100\r<BR>\n" );
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document.write( "0.25x+3.50y+10z = 100\r<BR>\n" );
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document.write( "Let's multiply both sides by 100 to clear out the decimals.<BR>\n" );
document.write( "0.25x+3.50y+10z = 100<BR>\n" );
document.write( "100*(0.25x+3.50y+10z) = 100*(100)<BR>\n" );
document.write( "25x+350y+1000z = 10000\r<BR>\n" );
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document.write( "So far we have these two equations<BR>\n" );
document.write( "x+y+z = 100<BR>\n" );
document.write( "25x+350y+1000z = 10000\r<BR>\n" );
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document.write( "I'll multiply everything in equation 1 by 25<BR>\n" );
document.write( "x+y+z = 100<BR>\n" );
document.write( "25(x+y+z) = 25(100)<BR>\n" );
document.write( "25x+25y+25z = 2500\r<BR>\n" );
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document.write( "So we have these two equations now<BR>\n" );
document.write( "25x+25y+25z = 2500<BR>\n" );
document.write( "25x+350y+1000z = 10000\r<BR>\n" );
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document.write( "Subtract straight down. The x terms will cancel out.<BR>\n" );
document.write( "25x-25x goes to 0x<BR>\n" );
document.write( "25y-350y goes to -325y<BR>\n" );
document.write( "25z-1000z goes to -975z<BR>\n" );
document.write( "2500-10000 goes to -7500\r<BR>\n" );
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document.write( "This is the new equation we have now<BR>\n" );
document.write( "-325y-975z = -7500\r<BR>\n" );
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document.write( "To clean things up a bit, we can divide both sides by -25<BR>\n" );
document.write( "-325y-975z = -7500<BR>\n" );
document.write( "(-325y-975z)/(-25) = -7500/(-25)<BR>\n" );
document.write( "(-325y)/(-25)-(975z)/(-25) = -7500/(-25)<BR>\n" );
document.write( "13y+39z = 300\r<BR>\n" );
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document.write( "Notice how the left side expression 13y+39z factors to 13(y+3z) showing that 13 is a factor of the left hand side.<BR>\n" );
document.write( "However, 13 is not a factor of 300 because 300/13 = 23.0769 approximately.\r<BR>\n" );
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document.write( "What this means is that the linear diophantine equation<BR>\n" );
document.write( "13y+39z = 300<BR>\n" );
document.write( "does <u>not</u> have any integer solutions for y & z.<BR>\n" );
document.write( "Consequently, it means the original system we started with doesn't have any nonnegative integer solutions for x, y and z.\r<BR>\n" );
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document.write( "<font color=red>Therefore, it is impossible to buy 100 animals with $100 based on these prices.</font><BR>\n" );
document.write( "Based on the phrasing, I'm assuming the goal is to have the farmer use up the entire $100 without any change leftover (otherwise there are many solutions possible).\r<BR>\n" );
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document.write( "Side note: <BR>\n" );
document.write( "I used computer software to check, just to be sure. I made 3 nested arrays loop through from 0 to 100 (for each variable x,y,z). <BR>\n" );
document.write( "That means there are 101^3 = 1,030,301 combos to check. An insane amount by hand. But to a computer, that's practically nothing.<BR>\n" );
document.write( "For each iteration, the software plugged in the particular x,y,z values into the two original equations mentioned. <BR>\n" );
document.write( "As expected, the program found no integer solutions. <BR>\n" );
document.write( "I wasn't sure how to show this aspect, but if you're curious then I'll share the source code.\r<BR>\n" );
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document.write( "Further thoughts:<BR>\n" );
document.write( "Here are some near misses. <BR>\n" );
document.write( "Each (x,y,z) triple satisfies x+y+z = 100, but the second equation isn't satisfied. <BR>\n" );
document.write( "The closest I could get to $100, while staying under $100, is $99.75<BR>\n" );
document.write( "Each of the triples shown below produce that final cost<BR>\n" );
document.write( "(x,y,z) = (77,23,0)<BR>\n" );
document.write( "(x,y,z) = (79,20,1)<BR>\n" );
document.write( "(x,y,z) = (81,17,2)<BR>\n" );
document.write( "(x,y,z) = (83,14,3)<BR>\n" );
document.write( "(x,y,z) = (85,11,4)<BR>\n" );
document.write( "(x,y,z) = (87,8,5)<BR>\n" );
document.write( "(x,y,z) = (89,5,6)<BR>\n" );
document.write( "(x,y,z) = (91,2,7)<BR>\n" );
document.write( "For example, trying the second triple mentioned gets us<BR>\n" );
document.write( "x+y+z = 79+20+1 = 100<BR>\n" );
document.write( "0.25x+3.5y+10z = 0.25*79+3.5*20+10*1 = 99.75<BR>\n" );
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