document.write( "Question 65472: This is a word problem question that has to be solved using cramers rule\r
\n" ); document.write( "\n" ); document.write( "At a college basketball game, student tickets cost 12 dollars each, adult tickets cost 16 dollars each and childrens tickets cost 7 dollars each. Four times as many adult tickets as children tickets were sold, the total number of childrens tickets and adult tickets was half the number of student tickets. The total number of tickets sales was 33,043 dollars. How many of each type of ticket was sold? \n" ); document.write( "
Algebra.Com's Answer #46029 by Edwin McCravy(20054) $\"\"$ $\"About$
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document.write( "This is a word problem question that has to be solved using cramers rule\r\n" );
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document.write( "At a college basketball game, student tickets cost 12 dollars each, \r\n" );
document.write( "adult tickets cost 16 dollars each and childrens tickets cost 7 dollars\r\n" );
document.write( "each. Four times as many adult tickets as children tickets were sold, \r\n" );
document.write( "the total number of childrens tickets and adult tickets was half the \r\n" );
document.write( "number of student tickets. The total  number of tickets sales was 33,043\r\n" );
document.write( "dollars. How many of each type of ticket was sold?\r\n" );
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document.write( "Let x = the number of student tickets\r\n" );
document.write( "Let y = the number of adult tickets\r\n" );
document.write( "Let z = the number of children's tickets\r\n" );
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document.write( ">>...Four times as many adult tickets as children tickets were sold,..<< \r\n" );
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document.write( "y = 4z\r\n" );
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document.write( ">>...the total number of childrens tickets and adult tickets was half\r\n" );
document.write( "the number of student tickets...<<\r\n" );
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document.write( "z + y = (1/2)S\r\n" );
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document.write( ">>...student tickets cost 12 dollars each, adult tickets cost 16 \r\n" );
document.write( "dollars each and childrens tickets cost 7 dollars each...<<\r\n" );
document.write( ">>...The total number of tickets sales was 33,043 dollars...<< \r\n" );
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document.write( "12x + 16y + 7z = 33043\r\n" );
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document.write( "So you have the system:\r\n" );
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document.write( "y = 4z\r\n" );
document.write( "z + y = (1/2)S\r\n" );
document.write( "12x + 16y + 7z = 33043\r\n" );
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document.write( "Now we have to rewrite the system so\r\n" );
document.write( "it can be solved by Cramer's rule.\r\n" );
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document.write( "Rewrite the first equation as\r\n" );
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document.write( "0x + 1y - 4z = 0\r\n" );
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document.write( "Clear the second of fractions by multiplying\r\n" );
document.write( "through by 2\r\n" );
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document.write( "2z + 2y = x\r\n" );
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document.write( "then rewrite as\r\n" );
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document.write( "-1x + 2y + 2z = 0\r\n" );
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document.write( "Leave the third one as it is, namely\r\n" );
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document.write( "12x + 16y + 7z = 33043\r\n" );
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document.write( "So the system we have to solve is\r\n" );
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document.write( " 0x +  1y - 4z =     0 \r\n" );
document.write( "-1x +  2y + 2z =     0\r\n" );
document.write( "12x + 16y + 7z = 33043\r\n" );
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document.write( "The system  must be lined up like the\r\n" );
document.write( "above to be solveable by Cramer's rule.\r\n" );
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document.write( "Form 4 determinants, D, D_x, D_y, and D_z\r\n" );
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document.write( "To form D, just write all the coefficients\r\n" );
document.write( "down left of the equal signs, like this:\r\n" );
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document.write( "    | 0   1  -4| \r\n" );
document.write( "D = |-1   2   2| \r\n" );
document.write( "    |12  16   7| \r\n" );
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document.write( "Now we will proceed to construct D_x, D_y\r\n" );
document.write( "and D_z from D\r\n" );
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document.write( "Notice that D does not contain the column\r\n" );
document.write( "of constants to the right of the equal sign\r\n" );
document.write( "in the system, which is:\r\n" );
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document.write( "    0\r\n" );
document.write( "    0\r\n" );
document.write( "33043\r\n" );
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document.write( "but D_x, D_y, and D_z will all contain it.\r\n" );
document.write( "Let's call this column the \"column of\r\n" );
document.write( "constants\".\r\n" );
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document.write( "x is the FIRST unknown, so replace only the\r\n" );
document.write( "FIRST column of D with the column of constants,\r\n" );
document.write( "and we have\r\n" );
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document.write( "     |    0   1  -4| \r\n" );
document.write( "D_x = |    0   2   2| \r\n" );
document.write( "     |33043  16   7| \r\n" );
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document.write( "y is the SECOND unknown, so replace only the\r\n" );
document.write( "SECOND column of D with the column of constants,\r\n" );
document.write( "and we have\r\n" );
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document.write( "     | 0      0  -4| \r\n" );
document.write( "D_y = |-1      0   2| \r\n" );
document.write( "     |12  33043   7|\r\n" );
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document.write( "z is the THIRD unknown, so replace only the\r\n" );
document.write( "THIRD column of D with the column of constants,\r\n" );
document.write( "and we have\r\n" );
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document.write( "     | 0   1      0| \r\n" );
document.write( "D_z = |-1   2      0| \r\n" );
document.write( "     |12  16  33043|\r\n" );
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document.write( "Do you know how to eveluate 3 by 3 determinants?\r\n" );
document.write( "If you don't repost asking how to eveluate a\r\n" );
document.write( "3 by 3 determinant and we'll show you how.\r\n" );
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document.write( "I will assume you already know how.\r\n" );
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document.write( "    | 0   1  -4| \r\n" );
document.write( "D = |-1   2   2| = 191 \r\n" );
document.write( "    |12  16   7|\r\n" );
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document.write( "     |    0   1  -4| \r\n" );
document.write( "D_x = |    0   2   2| = 330430 \r\n" );
document.write( "     |33043  16   7|\r\n" );
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document.write( "     | 0      0  -4| \r\n" );
document.write( "D_y = |-1      0   2| = 132172 \r\n" );
document.write( "     |12  33043   7|\r\n" );
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document.write( "     | 0   1      0| \r\n" );
document.write( "D_z = |-1   2      0| = 33043 \r\n" );
document.write( "     |12  16  33043|\r\n" );
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document.write( "Now the formulas for x, y and z are\r\n" );
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document.write( "x = D_x/D, y = D_y/D, and z = D_z/D\r\n" );
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document.write( "--------------------------------\r\n" );
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document.write( "x = D_x/D = 330430/191 = 1730\r\n" );
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document.write( "y = D_y/D = 132172/191 = 692\r\n" );
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document.write( "z = D_z/D = 33043/192 = 173\r\n" );
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document.write( "Edwin
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