document.write( "Question 850826: please help me solve this problem: Find the number of non-negative solution of 3x+7y=999 \n" ); document.write( "
Algebra.Com's Answer #512454 by Edwin McCravy(20055)\"\" \"About 
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document.write( "Fombitz thought you meant all solutions, but you probably meant \r\n" );
document.write( "only solutions (x,y) where x and y are both whole numbers:\r\n" );
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document.write( "3x + 7y = 999\r\n" );
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document.write( "Write all the numbers in terms of the nearest multiple to the absolute\r\n" );
document.write( "value of the coefficient of the variable with the smallest absolute \r\n" );
document.write( "value that are not already a multiple of it:\r\n" );
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document.write( "Of the numbers 3, 7, 333, only  7 is not a multiple of 3, so we write \r\n" );
document.write( "7 in terms of its nearest multiple of 3, which is 6. So we write\r\n" );
document.write( "7 as 6+1\r\n" );
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document.write( "3x+(6+1)y = 999\r\n" );
document.write( "  3x+6y+y = 999\r\n" );
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document.write( "WE divide every term by 3\r\n" );
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document.write( "  x+2y+y/3 = 333\r\n" );
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document.write( "y/3 = 333-x\r\n" );
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document.write( "Since 333-x is a non-negative integer, so is y/3.\r\n" );
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document.write( "Let the integer be A, then\r\n" );
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document.write( "333-x = A  and y/3 = A\r\n" );
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document.write( "So y = 3A\r\n" );
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document.write( "We substitute that in\r\n" );
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document.write( "3x + 7y = 999\r\n" );
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document.write( "3x + 7(3A) = 999\r\n" );
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document.write( "3x + 21A = 999\r\n" );
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document.write( "Divide through by 3\r\n" );
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document.write( "x + 7A = 333\r\n" );
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document.write( "x = 333-7A\r\n" );
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document.write( "Now since x >= 0,  \r\n" );
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document.write( "333-7A >= 0\r\n" );
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document.write( "-7A >= -333\r\n" );
document.write( "  A <= 333/7 = 47 4/7\r\n" );
document.write( "  A <= 47\r\n" );
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document.write( "Also since y >= 0\r\n" );
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document.write( "  3A >= 0 \r\n" );
document.write( "   A >= 0\r\n" );
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document.write( "So there are 48 solutions from A = 0 through 47 inclusively.\r\n" );
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document.write( "Edwin
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