document.write( "Question 253759: you have 20 dollars. you have to buy roosters, hens and chicks. roosters cost $3.00 a piece, $1.00 for hens, and $0.50 for chicks. you must buy 20 heads. how many for each?\r
\n" ); document.write( "\n" ); document.write( "i have 3 unknowns in here but i only can think of 2 equations:\r
\n" ); document.write( "\n" ); document.write( "20 = x+y+z
\n" ); document.write( "20 = 3x+y+0.5z \r
\n" ); document.write( "\n" ); document.write( "please help me solve this problem\r
\n" ); document.write( "\n" ); document.write( "thanks
\n" ); document.write( "adrian \n" ); document.write( "

Algebra.Com's Answer #186083 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This is called an \"over-determined system,\r\n" );
document.write( "so there are likely more than one solution.\r\n" );
document.write( "\r\n" );
document.write( "Eliminate y and get\r\n" );
document.write( "\r\n" );
document.write( "4x = z\r\n" );
document.write( "\r\n" );
document.write( "Eliminate x and get\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "4y = 80-5z\r\n" );
document.write( "\r\n" );
document.write( "4y = 80-(4+1)z\r\n" );
document.write( "\r\n" );
document.write( "4y = 80-4z-z\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 4\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Isolate the fraction:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The right side is an integer, so the right side is too\r\n" );
document.write( "\r\n" );
document.write( "Let that intger be A\r\n" );
document.write( "\r\n" );
document.write( "then  and 20-z-y = A\r\n" );
document.write( "\r\n" );
document.write( "z = 4A\r\n" );
document.write( "\r\n" );
document.write( "Substitute in\r\n" );
document.write( "\r\n" );
document.write( "4y = 80-5z\r\n" );
document.write( "\r\n" );
document.write( "4y = 80-5(4A)\r\n" );
document.write( "\r\n" );
document.write( "4y = 80-20A\r\n" );
document.write( "\r\n" );
document.write( "y = 20-5A\r\n" );
document.write( " \r\n" );
document.write( "Substitute that and z = 4A in\r\n" );
document.write( "\r\n" );
document.write( "20 = x+y+z\r\n" );
document.write( "\r\n" );
document.write( "20 = x+(20-5A)+4A\r\n" );
document.write( "20 = x+20-5A+4A\r\n" );
document.write( "20 = x+20-A\r\n" );
document.write( "-x = -A\r\n" );
document.write( " x = A\r\n" );
document.write( "\r\n" );
document.write( "So we have\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "All must be > 0\r\n" );
document.write( "\r\n" );
document.write( "so\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Solving the first and third give A > 0\r\n" );
document.write( "\r\n" );
document.write( "Solving the second one:\r\n" );
document.write( "\r\n" );
document.write( "20-5A > 0\r\n" );
document.write( "  -5A > -20\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by -5 reverses the inequality\r\n" );
document.write( "\r\n" );
document.write( "    A < 4\r\n" );
document.write( "\r\n" );
document.write( "0 < A < 4\r\n" );
document.write( "  \r\n" );
document.write( "So A is in the set {1, 2, 3}\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "with A=1 becomes\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "with A=2 \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "with A=3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The problem has 3 solutions.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );