document.write( "Question 935888: determine the value (s)of a & b so that the system
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Algebra.Com's Answer #569263 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Lene them up vertically so that the letters are in \r\n" );
document.write( "separate columns:\r\n" );
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document.write( " x - 2y + bz = 3\r\n" );
document.write( "ax      + 2z = 2\r\n" );
document.write( "5x + 2y      = 1\r\n" );
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document.write( "Since y is eliminated from the middle equation,\r\n" );
document.write( "let's eliminate y from the 1st and 3rd equations.\r\n" );
document.write( "We can do that simply by adding them\r\n" );
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document.write( " x - 2y + bz = 3\r\n" );
document.write( "5x + 2y      = 1\r\n" );
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document.write( "6x      + bz = 4\r\n" );
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document.write( "Now we have this system:\r\n" );
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document.write( "     ax + 2z = 2\r\n" );
document.write( "     6x + bz = 4\r\n" );
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document.write( "Let's eliminate either x or z.  I'll pick z\r\n" );
document.write( "to eliminate:\r\n" );
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document.write( "Multiply the first by b and the second by -2\r\n" );
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document.write( "     abx + 2bz = 2b\r\n" );
document.write( "    -12x - 2bz = -8\r\n" );
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document.write( "(ab-12)x       = 2b-8\r\n" );
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document.write( "There are three cases:\r\n" );
document.write( "1. The left side is 0 and the right side is not 0. There is no solution.\r\n" );
document.write( "2. The left side is not 0. There can only be on value of x, and hence one value\r\n" );
document.write( "   for y and z as well.\r\n" );
document.write( "3. Both sides are 0. Then x can be any value and there can be infinitely many\r\n" );
document.write( "   solutions. \r\n" );
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document.write( "The coefficient of x is 0 when and only when ab-12=0 or ab=-12\r\n" );
document.write( "The right side is 0 when and only when 2b-8=0, or b=4.\r\n" );
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document.write( "1. If the right side is NOT 0, then there can be no solution,\r\n" );
document.write( "because a left side of 0 cannot equal to a right side which\r\n" );
document.write( "is something other than 0.\r\n" );
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document.write( "2. If the coefficient of x is not 0, there is a unique solution,\r\n" );
document.write( "for then \"x=%282b-a%29%2F%28ab-12%29\" which will have a single value\r\n" );
document.write( "for the denominator will not be 0.\r\n" );
document.write( "So there is a unique solution if  ab ≠ 12\r\n" );
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document.write( "3.  There are infinitely many solutions if both the coefficient of\r\n" );
document.write( "x and the right side are both 0.  That's because x can be any \r\n" );
document.write( "number whatever and 0x = 0 will always be true.  That will be\r\n" );
document.write( "when ab = 12 and b = 4.\r\n" );
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document.write( "Answer:\r\n" );
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document.write( "1. There are no solutions if  ab = 12 and b ≠ 4\r\n" );
document.write( "2. There is one unique solution if  ab ≠ 12 \r\n" );
document.write( "3. There are infinitely many solutions if ab = 12 and b = 4\r\n" );
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document.write( "Edwin
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