document.write( "Question 651693: for what values of k is the system of equations kx+3y=k-2,12x+ky=k inconsistent . \n" ); document.write( "
Algebra.Com's Answer #407785 by Edwin McCravy(20059)\"\" \"About 
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document.write( "The system of equations:\r\n" );
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document.write( "\"system%28Ax%2BBy=C%2CDx%2BEy=F%29\"\r\n" );
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document.write( "is \r\n" );
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document.write( "(1)  consistent and independent if \r\n" );
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document.write( "AE ≠ BD \r\n" );
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document.write( "(2)  consistent and dependent if \r\n" );
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document.write( "AE = BD and AF = CD\r\n" );
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document.write( "\"system%28kx%2B3y=k-2%2C12x%2Bky=k%29\"\r\n" );
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document.write( "Here A=k, B=3, C=k-2, D=12, E=k, F=k\r\n" );
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document.write( "Since we want them to be inconsistent, we must rule out \r\n" );
document.write( "both case (1) and case (2).\r\n" );
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document.write( "To rule out case (1) we require AE ≠ BD to be false, so we must\r\n" );
document.write( "require:\r\n" );
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document.write( "  AE = BD\r\n" );
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document.write( "k(k) = 3(12)\r\n" );
document.write( "  kČ = 36\r\n" );
document.write( "   k = ±6\r\n" );
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document.write( "But we also must check to see that k = ±6 also rules out \r\n" );
document.write( "case (2)\r\n" );
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document.write( "That is we must be sure that AF ≠ CD\r\n" );
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document.write( "    AF ≠ CD\r\n" );
document.write( "  k(k) ≠ (k-2)(12)\r\n" );
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document.write( "±6(±6) ≠ (±6-2)(12)\r\n" );
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document.write( "Taking the + we have\r\n" );
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document.write( "   36 ≠ (6-2)(12)\r\n" );
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document.write( "   36 ≠ (4)(12)\r\n" );
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document.write( "   36 ≠ 48\r\n" );
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document.write( "which is true.\r\n" );
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document.write( "Taking the - we have\r\n" );
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document.write( "   36 ≠ (-6-2)(12)\r\n" );
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document.write( "   36 ≠ (-8)(12)\r\n" );
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document.write( "   36 ≠ -96\r\n" );
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document.write( "which is also true.\r\n" );
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document.write( "So case (2) is ruled out be taking k = ±6\r\n" );
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document.write( "It was necessary to rule out case (2), however.\r\n" );
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document.write( "Edwin
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