document.write( "Question 1121649: In a class of 40 students, 17 have ridden an airplane, 28 have ridden a boat, 10 have ridden a train, 12 have ridden both an airplane and a boat. 3 have ridden a train only and 4 have ridden an airplane only. Some students have not yet ridden any of the three modes of transportation and an equal number have taken all the three. How many students have used all the three modes of transportation? How many students have taken only the boat? \n" ); document.write( "
Algebra.Com's Answer #737634 by Edwin McCravy(20056)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "In a class of 40 students,
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document.write( "That means:\r\n" );
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document.write( "(1) c+d+e+f+g+h+i+j = 40\r\n" );
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\n" ); document.write( "17 have ridden an airplane,
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document.write( "(2) c+d+f+g = 17\r\n" );
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\n" ); document.write( "28 have ridden a boat,
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document.write( "(3) d+e+g+h = 28\r\n" );
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\n" ); document.write( "10 have ridden a train,
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document.write( "f+g+h+i = 10\r\n" );
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\n" ); document.write( "12 have ridden both an airplane and a boat.
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document.write( "(4) d+g = 12\r\n" );
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\n" ); document.write( "3 have ridden a train only
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document.write( "(5) i = 3\r\n" );
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\n" ); document.write( "and 4 have ridden an airplane only.
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document.write( "(6) c = 4\r\n" );
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\n" ); document.write( "Some students have not yet ridden any of
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document.write( "(7) j > 0 (that is, j is not 0)\r\n" );
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\n" ); document.write( "and an equal number have taken all the three.
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document.write( "(8) g = j  but we'll need the letters on the left side\r\n" );
document.write( "so we'll write it as:\r\n" );
document.write( "(8) g-j = 0\r\n" );
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\n" ); document.write( "How many students have used all the three
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document.write( "This asks for the value of g.\r\n" );
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\n" ); document.write( "How many students have taken only the boat?
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document.write( "This asks for the value of e.\r\n" );
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document.write( "So we put all equations down and line up the unknowns\r\n" );
document.write( "[we don't need (7)]\r\n" );
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document.write( "(1) c+d+e+f+g+h+i+j = 40\r\n" );
document.write( "(2) c+d  +f+g       = 17\r\n" );
document.write( "(3)   d+e  +g+h     = 28\r\n" );
document.write( "(4)   d    +g       = 12\r\n" );
document.write( "(5)             i   =  3\r\n" );
document.write( "(6) c               =  4\r\n" );
document.write( "(8)         g       =  j\r\n" );
document.write( "\r\n" );
document.write( "Change the bottom equation to g-j = 0\r\n" );
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document.write( "(1) c+d+e+f+g+h+i+j = 40\r\n" );
document.write( "(2) c+d  +f+g       = 17\r\n" );
document.write( "(3)   d+e  +g+h     = 28\r\n" );
document.write( "(4)   d    +g       = 12\r\n" );
document.write( "(5)             i   =  3\r\n" );
document.write( "(6) c               =  4\r\n" );
document.write( "(8)         g    -j =  0\r\n" );
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document.write( "We have the 7 x 9 augmented matrix:\r\n" );
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document.write( "Using row operations to reduce this matrix to\r\n" );
document.write( "row reduced echelon form, we have:\r\n" );
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document.write( "Which describes this system:\r\n" );
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document.write( "c = 4\r\n" );
document.write( "d = 8\r\n" );
document.write( "e = 16\r\n" );
document.write( "f = 1\r\n" );
document.write( "g = 4\r\n" );
document.write( "i = 3\r\n" );
document.write( "j = 4\r\n" );
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document.write( "So we have the answers g = 4 and e = 16.\r\n" );
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document.write( "But let's complete the Venn diagram:\r\n" );
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document.write( "We have everything but h.\r\n" );
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document.write( "But we can find h by substituting in\r\n" );
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document.write( "(3)   d+e  +g+h     = 28\r\n" );
document.write( "      8+16 +4+h     = 28\r\n" );
document.write( "           28+h     = 28\r\n" );
document.write( "              h     = 0\r\n" );
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document.write( "So the complete Venn diagram is:\r\n" );
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document.write( "Edwin
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