document.write( "Question 697720: At some stops, a certain bus picks up 5 people. At other stops, it picks up 2 and, at the same time lets off 5. There are no other stops but these. It starts its run empty and picks up 5 people. At the end, it has 11 people aboard.\r
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Algebra.Com's Answer #430270 by Edwin McCravy(20060)\"\" \"About 
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At some stops, a certain bus picks up 5 people. At other stops, it picks up 2 and, at the same time lets off 5.
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document.write( "Let x = the number of times the bus increases the no. of passengers by 5.\r\n" );
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document.write( "When the bus picks up 2 and lets off 5, it decreases the no. of passengers\r\n" );
document.write( "by 3.\r\n" );
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document.write( "Let y = the number of times the bus decreases the no. of passengers by 3.\r\n" );
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document.write( "Then the number of passengers is 5x - 3y \r\n" );
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\n" ); document.write( "There are no other stops but these. It starts its run empty and picks up 5 people. At the end, it has 11 people aboard.
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document.write( "So we have\r\n" );
document.write( "      5x - 3y = 11\r\n" );
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document.write( "The smallest coefficient in absolute value is |-3| = 3.\r\n" );
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document.write( "So we write 5 and ll in terms of their nearest multiple of 3.\r\n" );
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document.write( "The nearest multiple of 3 to 5 is 6, so we write 5 as 6-1,\r\n" );
document.write( "The nearest multiple of 3 to 11 is 12, so we write 11 as 12-1, \r\n" );
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document.write( "  (6-1)x - 3y = 12-1\r\n" );
document.write( "  6x - x - 3y = 12 - 1\r\n" );
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document.write( "Divide every term through by 3\r\n" );
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document.write( "  2x - \"x%2F3\" - y = 4 - \"1%2F3\"\r\n" );
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document.write( "Isolate all the fractions on the right  \r\n" );
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document.write( " 2x - y - 4 = \"x%2F3\" - \"1%2F3\"\r\n" );
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document.write( "The left side is an integer, so the right side is also an\r\n" );
document.write( "integer.  Let that integer be A.  Then\r\n" );
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document.write( " 2x - y - 4 = A  and  \"x%2F3\" - \"1%2F3\" = A\r\n" );
document.write( "                      x - 1 = 3A\r\n" );
document.write( "                          x = 3A + 1\r\n" );
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document.write( "Substitute 3A+1 for x in the left equation\r\n" );
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document.write( "2(3A+1) - y - 4 = A\r\n" );
document.write( " 6A + 2 - y - 4 = A\r\n" );
document.write( "     6A - 2 - y = A\r\n" );
document.write( "         5A - 2 = y\r\n" );
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document.write( "So we have (x,y) = (3A+1,5A-2)\r\n" );
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\n" ); document.write( "a. If the number of stops is greater than 75, what is the least number of stops the bus makes?
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document.write( "          x + y > 75\r\n" );
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document.write( "Substitute (x,y) = (3A+1,5A-2)\r\n" );
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document.write( "    3A+1 + 5A-2 > 75\r\n" );
document.write( "           8A-1 > 75\r\n" );
document.write( "             8A > 76\r\n" );
document.write( "              A > \"76%2F8\"\r\n" );
document.write( "              A > 9.5\r\n" );
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document.write( "x + y increases with increasing A, so the\r\n" );
document.write( "minimum value of x+y (the number of stops, is when A\r\n" );
document.write( "is the smallest integer greater than 9.5, which is 10.\r\n" );
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document.write( "Therefore (x,y) = (3A+1,5A-2) = (3·10+1,5·10-2) = (31,48)\r\n" );
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document.write( "So the smallest number of stops is the minimum value of x+y = 31+48 = 79\r\n" );
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\n" ); document.write( "b. If the number of stops is greater than 703, what is the least number of stops the bus makes?
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document.write( "          x + y > 703\r\n" );
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document.write( "Substitute (x,y) = (3A+1,5A-2)\r\n" );
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document.write( "    3A+1 + 5A-2 > 703\r\n" );
document.write( "           8A-1 > 703\r\n" );
document.write( "             8A > 704\r\n" );
document.write( "              A > \"704%2F8\"\r\n" );
document.write( "              A > 88\r\n" );
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document.write( "x + y increases with increasing A, so the\r\n" );
document.write( "minimum value of x+y (the number of stops), is when A\r\n" );
document.write( "is the smallest integer greater than 88, which is 89.\r\n" );
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document.write( "Therefore (x,y) = (3A+1,5A-2) = (3·89+1,5·89-2) = (268,443)\r\n" );
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document.write( "So the smallest number of stops is the minimum value of x+y = 268+443 = 711\r\n" );
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\n" ); document.write( "c. For part A, how many of the stops must be where the total gain in passengers is 5?
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document.write( "That's just the value of x for part a, which is 31.\r\n" );
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\n" ); document.write( "For part B?
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document.write( "That's just the value of x for part b, which is 268.\r\n" );
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document.write( "Edwin
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