document.write( "Question 79410: solve using cramers rule:\r
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\n" ); document.write( "3x+1y-1z=-5
\n" ); document.write( "5x-2y-4z=32\r
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\n" ); document.write( "\n" ); document.write( "I am very confused with this problem...please help! \n" ); document.write( "
Algebra.Com's Answer #57031 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "solve using cramers rule: \r\n" );
document.write( "2x-3y-3z=25\r\n" );
document.write( "3x+1y-1z=-5\r\n" );
document.write( "5x-2y-4z=32 \r\n" );
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document.write( "I am very confused with this problem...please help!\r\n" );
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document.write( "Cramer's rule can only be used to solve independent\r\n" );
document.write( "systems.  If the system is dependent or inconsistent,\r\n" );
document.write( "then Cramer's rule cannot be used to solve the system.\r\n" );
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document.write( "There are 4 determinants that must be made with Cramer's \r\n" );
document.write( "rule: D, Dx, Dy, and Dz.\r\n" );
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document.write( "If it turns out that D is not 0, then \r\n" );
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document.write( "x = Dx/D, y = Dy/D, and z = Dz/D.\r\n" );
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document.write( "However if it turns out that D = 0, then Cramer's rule \r\n" );
document.write( "cannot be used, because either the system is dependent \r\n" );
document.write( "and has INFINITELY MANY solutions or else it is \r\n" );
document.write( "inconsistent and has NO solutions.\r\n" );
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document.write( "First we make D. Here is how:\r\n" );
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document.write( "Start with the system:\r\n" );
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document.write( "2x-3y-3z=25\r\n" );
document.write( "3x+1y-1z=-5\r\n" );
document.write( "5x-2y-4z=32\r\n" );
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document.write( "Erase all the letters, the equal signs, and the\r\n" );
document.write( "numbers on the right of the equal sign\r\n" );
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document.write( "2 -3 -3 \r\n" );
document.write( "3 +1 -1 \r\n" );
document.write( "5 -2 -4 \r\n" );
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document.write( "Put bars around it:\r\n" );
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document.write( "|2 -3 -3| \r\n" );
document.write( "|3 +1 -1| \r\n" );
document.write( "|5 -2 -4|\r\n" );
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document.write( "That's the determinant called D.\r\n" );
document.write( "Do you know how to evaluate a 3x3 determinant?  If\r\n" );
document.write( "you don't post again and ask how to evaluate it. I\r\n" );
document.write( "will assume you know how.\r\n" );
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document.write( "It turns out that D = 0, so Cramer's rule cannot be\r\n" );
document.write( "used to solve this system. The system is either\r\n" );
document.write( "dependent and has INFINITELT MANY solutions, or it\r\n" );
document.write( "is inconsistent and has NO solutions.  To find out\r\n" );
document.write( "which it is, we form Dx, Dy and Dz.  If they are ALL\r\n" );
document.write( "0, then the system is dependent and has infinitely\r\n" );
document.write( "many solutions.  If ANY ONE of Dx, Dy, or Dz is NOT 0,\r\n" );
document.write( "then the system is inconsistent and there are NO solutions.\r\n" );
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document.write( "x is the FIRST unknown that appears, so \r\n" );
document.write( "Dx is just like D except that the FIRST \r\n" );
document.write( "column is replaced by the column of \r\n" );
document.write( "constants.  These are the numbers on the \r\n" );
document.write( "right of the equal signs in the system, \r\n" );
document.write( "i.e., the red numbers below:\r\n" );
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document.write( "2x-3y-3z=25\r\n" );
document.write( "3x+1y-1z=-5\r\n" );
document.write( "5x-2y-4z=32\r\n" );
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document.write( "Here is Dx:\r\n" );
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document.write( "|25 -3 -3|\r\n" );
document.write( "|-5 +1 -1|\r\n" );
document.write( "|32 -2 -4|\r\n" );
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document.write( "y is the SECOND unknown that appears, so \r\n" );
document.write( "Dy is just like D except that the SECOND \r\n" );
document.write( "column is replaced by the column of \r\n" );
document.write( "constants. \r\n" );
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document.write( "Here is Dy:\r\n" );
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document.write( "|2 25 -3|\r\n" );
document.write( "|3 -5 -1|\r\n" );
document.write( "|5 32 -4|\r\n" );
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document.write( "z is the THIRD unknown that appears, so \r\n" );
document.write( "Dz is just like D except that the THIRD \r\n" );
document.write( "column is replaced by the column of \r\n" );
document.write( "constants. \r\n" );
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document.write( "Here is Dz:\r\n" );
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document.write( "|2 -3 25|\r\n" );
document.write( "|3 +1 -5|\r\n" );
document.write( "|5 -2 32|\r\n" );
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document.write( "Evaluate those\r\n" );
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document.write( "Dx = 72,  Dy = -84,  Dz = 132\r\n" );
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document.write( "They certainly are NOT all 0.  In fact,\r\n" );
document.write( "none of them are, so the system is \r\n" );
document.write( "INCONSISTENT and there is no solution.\r\n" );
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document.write( "Edwin

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