document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197109>Question 1197109</A>: <I><SPAN STYLE=\"word-break: break-all;\"> vip seating reserved seating and general admission tickets were sold for the school play at $15 $10 and 5$ each respectively the drama department sold 360 tickets for a total of $2800 if there were 40 more general admission tickets than the total number of vip and reserved tickets how many of each type of ticket were sold </SPAN>\n" );
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document.write( "x = number of VIP tickets<BR>\n" );
document.write( "y = number of reserved seating tickets<BR>\n" );
document.write( "z = number of general admission tickets\r<BR>\n" );
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document.write( "x,y,z are nonnegative integers<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Ticket Type</TD><TD >Cost per ticket</TD><TD ># of tickets</TD><TD >Subtotal cost</TD></tr><tr><TD >VIP</TD><TD >15</TD><TD >x</TD><TD >15x</TD></tr><tr><TD >Reserved Seating</TD><TD >10</TD><TD >y</TD><TD >10y</TD></tr><tr><TD >General Admission</TD><TD >5</TD><TD >z</TD><TD >5z</TD></tr><tr><TD >Total</TD><TD ></TD><TD >360</TD><TD >2800</TD></tr>\n" );
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document.write( "The second to last column forms this equation<BR>\n" );
document.write( "x+y+z = 360<BR>\n" );
document.write( "this is to represent the idea adding the ticket counts gets to 360 tickets total\r<BR>\n" );
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document.write( "The last column forms the equation 15x+10y+5z = 2800<BR>\n" );
document.write( "15x = cost of just the VIP tickets (selling x of them for $15 each)<BR>\n" );
document.write( "10y = cost of just the reserved seating tickets<BR>\n" );
document.write( "5z = cost of just the general admission tickets\r<BR>\n" );
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document.write( "We're told that \"there were 40 more general admission tickets than the total number of vip and reserved tickets\"<BR>\n" );
document.write( "So we know that z = x+y+40\r<BR>\n" );
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document.write( "Think of it like saying<BR>\n" );
document.write( "(# of general admission) = ( (# of VIP) + (# of reserved seating) ) + 40\r<BR>\n" );
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document.write( "The system of equations is <BR>\n" );
document.write( "x+y+z = 360<BR>\n" );
document.write( "15x+10y+5z = 2800<BR>\n" );
document.write( "z = x+y+40\r<BR>\n" );
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document.write( "Because z = x+y+40, we can replace each z with x+y+40 in the first two equations<BR>\n" );
document.write( "Let's do so for the first equation<BR>\n" );
document.write( "x+y+z = 360<BR>\n" );
document.write( "x+y+x+y+40 = 360<BR>\n" );
document.write( "2x+2y+40 = 360<BR>\n" );
document.write( "2x+2y = 360-40<BR>\n" );
document.write( "2(x+y) = 320<BR>\n" );
document.write( "x+y = 320/2<BR>\n" );
document.write( "x+y = 160\r<BR>\n" );
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document.write( "and the second equation as well<BR>\n" );
document.write( "15x+10y+5z = 2800<BR>\n" );
document.write( "15x+10y+5(x+y+40) = 2800<BR>\n" );
document.write( "15x+10y+5x+5y+200 = 2800<BR>\n" );
document.write( "20x+15y+200 = 2800<BR>\n" );
document.write( "20x+15y = 2800-200<BR>\n" );
document.write( "5(4x+3y) = 2600<BR>\n" );
document.write( "4x+3y = 2600/5<BR>\n" );
document.write( "4x+3y = 520\r<BR>\n" );
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document.write( "We have this reduced system of equations<BR>\n" );
document.write( "x+y = 160<BR>\n" );
document.write( "4x+3y = 520\r<BR>\n" );
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document.write( "Let's solve the first equation for y<BR>\n" );
document.write( "x+y = 160<BR>\n" );
document.write( "y = 160-x\r<BR>\n" );
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document.write( "Then plug it into the second equation and solve for x.<BR>\n" );
document.write( "4x+3y = 520<BR>\n" );
document.write( "4x+3(160-x) = 520<BR>\n" );
document.write( "4x+480-3x = 520<BR>\n" );
document.write( "x+480 = 520<BR>\n" );
document.write( "x = 520-480<BR>\n" );
document.write( "x = 40<BR>\n" );
document.write( "There were 40 VIP tickets sold\r<BR>\n" );
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document.write( "Then we can determine y based on this x value.<BR>\n" );
document.write( "y = 160-x<BR>\n" );
document.write( "y = 160-40<BR>\n" );
document.write( "y = 120<BR>\n" );
document.write( "There were 120 reserved seat tickets sold.\r<BR>\n" );
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document.write( "Lastly let's compute z<BR>\n" );
document.write( "z = x+y+40<BR>\n" );
document.write( "z = 40+120+40<BR>\n" );
document.write( "z = 200<BR>\n" );
document.write( "There were 200 general admission tickets sold.\r<BR>\n" );
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document.write( "Here's what the table looks like after replacing x,y,z with 40,120, and 200 in that order.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Ticket Type</TD><TD >Cost per ticket</TD><TD ># of tickets</TD><TD >Subtotal cost</TD></tr><tr><TD >VIP</TD><TD >15</TD><TD >40</TD><TD >600</TD></tr><tr><TD >Reserved</TD><TD >10</TD><TD >120</TD><TD >1200</TD></tr><tr><TD >General Admission</TD><TD >5</TD><TD >200</TD><TD >1000</TD></tr><tr><TD >Total</TD><TD ></TD><TD >360</TD><TD >2800</TD></tr>\n" );
document.write( "</TABLE>Check:<BR>\n" );
document.write( "40+120+200 = 360<BR>\n" );
document.write( "600+1200+1000 = 2800<BR>\n" );
document.write( "200 general admission = (40 VIP + 120 reserved) + 40 = 40+120+40 = 200<BR>\n" );
document.write( "The answers are confirmed.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "40 VIP tickets<BR>\n" );
document.write( "120 reserved seating tickets<BR>\n" );
document.write( "200 general admission tickets<BR>\n" );
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