document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139813>Question 1139813</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An animal breeder can buy four types of food for Vietnamese​ pot-bellied pigs. Each case of Brand A contains 25 units of​ fiber, 30 units of​ protein, and 30 units of fat. Each case of Brand B contains 100 units of​ fiber, 80 units of​ protein, and 70 units of fat. Each case of Brand C contains 275 units of​ fiber, 210 units of​ protein, and 190 units of fat. Each case of Brand D contains 100 units of​ fiber, 80 units of​ protein, and 60 units of fat. How many cases of each brand should the breeder mix together to obtain a food that provides 3950 units of​ fiber, 3060 units of​ protein, and 2740 units of​ fat? \r<BR>\n" );
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document.write( "Let x represent the number of cases of Brand​ A, y represent the number of cases of Brand​ B, z represent the number of cases of Brand​ C, and w represent be the number of cases of Brand D.  There are four ways in which the breeder can mix brands to obtain a food that provides 3950 units of​ fiber, 3060 units of​ protein, and 2740 units of fat.\r<BR>\n" );
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document.write( "If w=​0, the solution is<BR>\n" );
document.write( "If w=​1, the solution is <BR>\n" );
document.write( "If w=​2, the solution is  </SPAN>\n" );
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document.write( "There are constraint equations for fiber, protein, and fat.<br><BR>\n" );
document.write( "fiber: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%2B100y%2B275z%2B100w+=+3950 ALIGN=MIDDLE  ALT=\"25x%2B100y%2B275z%2B100w+=+3950\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "protein: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30x%2B80y%2B210z%2B80w+=+3060 ALIGN=MIDDLE  ALT=\"30x%2B80y%2B210z%2B80w+=+3060\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "fat: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30x%2B70y%2B190z%2B60w+=+2740 ALIGN=MIDDLE  ALT=\"30x%2B70y%2B190z%2B60w+=+2740\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That's a system of three equation in four unknowns; it will have (if the problem is well formulated) a family of solutions in non-negative integers.<br><BR>\n" );
document.write( "Solving the system algebraically would be very tedious.  A solution using matrices on a graphing calculator produces the following set of equations relating w, x, y, and z:<br><BR>\n" );
document.write( "x-w = 0<BR>\n" );
document.write( "y+4w = 12<BR>\n" );
document.write( "z-w = 10<br><BR>\n" );
document.write( "Use those equations to express x, y, and z in terms of w; then substitute the values shown for w to find the different ways the breeder can get the right amounts of fiber, protein, and fat.<br><BR>\n" );
document.write( "Note: The problem as you state it says there are four ways to get the right amounts, but it seems to be asking for only the solutions where w is 0, 1, or 2.  There is in fact a fourth value of w that provides a solution. </SPAN>\n" );
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