document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194075>Question 1194075</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At some stops, a certain bus picks up 5 people. At other stops it picks up 2 and at the same time, let’s off 5. There are no other stops than these. It starts it’s run empty and picks up 5 people. At the end, it has 11 people aboard. If the number of stops is greater than 17, what is the least number of stops the bus makes? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826170 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "x = number of stops where the bus picks up 5 people<BR>\n" );
document.write( "y = number of stops where the bus picks up 2 people and lets off 5<br><BR>\n" );
document.write( "At each of the x stops, the number of people on the bus increases by 5.<BR>\n" );
document.write( "At each of the y stops, the number of people on the bus decreases by 3.<br><BR>\n" );
document.write( "Then after (x+y) stops the number of people on the bus is 5x-3y.<br><BR>\n" );
document.write( "We are to find the least number of total stops (x+y) greater than 17 for which the number of people left on the bus is 11:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-3y=11 ALIGN=MIDDLE  ALT=\"5x-3y=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; subject to the constraint that x+y>17<br><BR>\n" );
document.write( "This is a linear Diophantine equation -- a single equation with two variables, in which the values of the variables are integers.<br><BR>\n" );
document.write( "Here is a standard formal way for solving this kind of equation.<br><BR>\n" );
document.write( "(1) Solve the equation for one of the variables (it doesn't matter which)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-3y=11 ALIGN=MIDDLE  ALT=\"5x-3y=11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x=3y%2B11 ALIGN=MIDDLE  ALT=\"5x=3y%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Divide by 5, writing the result on the right as quotient plus remainder:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%283y%2B11%29%2F5+=+%2810%2B%283y%2B1%29%29%2F5+=+2%2B%283y%2B1%29%2F5 ALIGN=MIDDLE  ALT=\"x+=+%283y%2B11%29%2F5+=+%2810%2B%283y%2B1%29%29%2F5+=+2%2B%283y%2B1%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) Find the values of y that make that expression an integer; find the corresponding values of x, and find the value of x+y.  We are looking for the smallest value of x+y that is greater than 17.<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "    y   x=2+(3y+1)/5  x+y\r\n" );
document.write( "  ------------------------\r\n" );
document.write( "    3   2+2 = 4        7\r\n" );
document.write( "    8   2+5 = 7       15\r\n" );
document.write( "   13   2+8 = 10      23</PRE><BR>\n" );
document.write( "ANSWER: 23<br><BR>\n" );
document.write( "A note about solving linear Diophantine equations by this method....<br><BR>\n" );
document.write( "Note that in the table above, the values of x and y (and x+y) form arithmetic sequences. Specifically, the possible y values have a common difference of 5 and the possible x values have a common difference of 3.  The \"5\" and \"3\" are because of the equation x=(3y+1)/5.<br><BR>\n" );
document.write( "If you are solving a problem like this and your list of x and/or y values does not form an arithmetic sequence, then some of your calculations are incorrect.<br><BR>\n" );
document.write( "At the same time, knowing that these lists of values form arithmetic sequences allows you to find other solutions without searching for x or y values.  For example, in this problem, seeing that the first two possible y values are 3 and 8, you know that the common difference is 5, so the next possible y values are 13, 18, 23, ...; and since the first two possible x values are 4 and 7, you now that the next possible corresponding x values are 10, 13, 16....<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );