document.write( "Question 24538: Hello: I am having a hard time with this problem:\r
\n" ); document.write( "\n" ); document.write( "For which values of \"a\" does the following system have zero solutions? One solution? Infinitely many solutions?\r
\n" ); document.write( "\n" ); document.write( "x1 + x2 + x3 = 4
\n" ); document.write( "x3 = 2
\n" ); document.write( "(a^2 - 4)x3 = a - 2\r
\n" ); document.write( "\n" ); document.write( "I wasnt sure if I should substitute x3 into the third equation, then solve for a. I re-read the chapter section and there are not any examples. \r
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Algebra.Com's Answer #13063 by AnlytcPhil(1807)\"\" \"About 
You can put this solution on YOUR website!
Hello: I am having a hard time with this problem: \r\n" );
document.write( "For which values of \"a\" does the following system have zero solutions? One\r\n" );
document.write( "solution? Infinitely many solutions? \r\n" );
document.write( "x1 + x2 + x3 = 4\r\n" );
document.write( "x3 = 2\r\n" );
document.write( "(a^2 - 4)x3 = a - 2 \r\n" );
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document.write( "I wasnt sure if I should substitute x3 into the third equation, then solve for\r\n" );
document.write( "a. I re-read the chapter section and there are not any examples. \r\n" );
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document.write( "x1 + x2 +        x3 = 4\r\n" );
document.write( "                 x3 = 2\r\n" );
document.write( "           (aČ-4)x3 = a-2\r\n" );
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document.write( "Since x3 = 2, if we substitute 2 for x3 in the third equation we get\r\n" );
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document.write( "            (aČ-4)(2) = a-2\r\n" );
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document.write( "              2aČ - 8 = a - 2 \r\n" );
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document.write( "          2aČ - a - 6 = 0\r\n" );
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document.write( "      (2a + 3)(a - 2) = 0\r\n" );
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document.write( " 2a + 3 = 0.    |     a - 2 = 0\r\n" );
document.write( "                |         a = 2\r\n" );
document.write( "     2a = -3    |\r\n" );
document.write( "      a = -3/2  |\r\n" );
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document.write( "In order to have a solution at all, a has to be one of these.\r\n" );
document.write( "So if a is neither of these, there are no solutions.\r\n" );
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document.write( "If a = -3/2 the bottome equation comes out x3 = 2, identical with the\r\n" );
document.write( "second equation.\r\n" );
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document.write( "And if we substitute 2 for x3 in the first equation, we get\r\n" );
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document.write( "x1 + x2 + 2 = 4 or\r\n" );
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document.write( "x1 + x2 = 2 or\r\n" );
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document.write( "x1 = 2 - x2.\r\n" );
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document.write( "So there are infinitely many solutions in this case, because there are\r\n" );
document.write( "infinitely many choices for x2 and each one gives a different value for x1.\r\n" );
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document.write( "If a = 2 , the bottome equation becomes 0x3 = 0, so any value of x3 would\r\n" );
document.write( "satisfy that, but the middle equation tells us that x3 = 2.  So that still\r\n" );
document.write( "gives us an infinite number of solutions.\r\n" );
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document.write( "So a = -3/2 or a = 2 gives infinitely many solutions. Any other value of a\r\n" );
document.write( "gives no solutions.  There are no values of a which give just one solution.\r\n" );
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document.write( "Edwin
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