document.write( "Question 1128090: A Broadway theater has 400 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $70, main seats for $55, and balcony seats for $35. If all the seats are sold, the gross revenue to the theater is $20,400. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,600. How many are there of each kind of seat? there are ___ orchestra seats, ____main seats, and ___ balcony seats. \n" ); document.write( "

Algebra.Com's Answer #744626 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "A Broadway theater has 400 seats, divided into orchestra, main, and balcony seating.
\n" ); document.write( "Orchestra seats sell for $70, main seats for $55, and balcony seats for $35.
\n" ); document.write( "If all the seats are sold, the gross revenue to the theater is $20,400.
\n" ); document.write( "If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,600.
\n" ); document.write( "How many are there of each kind of seat?
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document.write( "The two key phrases in the condition are:\r\n" );
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document.write( "    \"If all the seats are sold, the gross revenue to the theater is $20,400\"\r\n" );
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document.write( "and \r\n" );
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document.write( "    \"If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,600\"\r\n" );
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document.write( "From these two statements, it follows immediately that half of the orchestra seats cost 20400 - 17600 = 2800 dollars;\r\n" );
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document.write( "hence,  all the orchestra seats cost twice of it, i.e. 5600 dollars.\r\n" );
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document.write( "Since the price of each single orchestra seat is $70, the number of the orchestra seats is   = 80.\r\n" );
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document.write( "In this way, we just reduced the problem from 3 unknown to only 2.\r\n" );
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document.write( "Let x be the number of the main seats and y be the number of the balcony seats.\r\n" );
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document.write( "Then you have this system of 2 equations\r\n" );
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document.write( "  x +   y = 400 - 80\r\n" );
document.write( "55x + 35y = 20400 - 5600\r\n" );
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document.write( "or, equivalently\r\n" );
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document.write( "  x +   y =   320,     (1)\r\n" );
document.write( "55x + 35y = 14800.     (2)\r\n" );
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document.write( "Use the elimination method. For it, multiply eq(1) by 55 (both sides), keeping eq(2) as is. You will get\r\n" );
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document.write( "55x + 55y = 55*320,     (1')\r\n" );
document.write( "55x + 35y = 14800.      (2')\r\n" );
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document.write( "Now subtract eq(2') from eq(1') to get\r\n" );
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document.write( "      20y = 55*320 - 14800  ====>  y =  = 140.\r\n" );
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document.write( "Then from eq(1),  x = 320 - 140 = 180.\r\n" );
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document.write( "ANSWER.  180 main seats;  140 balcony seats;  and  80 orchestra seats.\r\n" );
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document.write( "CHECK.   70*80 + 180*55 + 140*35 = 20400 dollars.    ! Correct !\r\n" );
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\n" ); document.write( "\n" ); document.write( "The lesson to learn from this solution:\r
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document.write( "    From the first glance, the problem is about 3 equations in 3 unknowns.\r\n" );
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document.write( "    But carefully reading, the student can find the hidden way to reduce it to the system of only 2 equations in 2 unknowns,\r\n" );
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document.write( "    which is much easier to solve.\r\n" );
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