Algebra.Com's Answer #847641 by ikleyn(52787)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "For certain values of k and m, the system\r
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document.write( "a + 2b = -3
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document.write( "4a + 2b = k - 2a - mb + 6a + 2b + 5\r
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document.write( "has infinitely many solutions (a,b). What are k and m?
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document.write( "Simplify second equation by combining common terms.\r\n" );
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document.write( "You will get an equivalent equation\r\n" );
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document.write( " 0 = k - mb + 5. (*)\r\n" );
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document.write( "If m =/= 0, then this equation has a unique solution for \"b\"\r\n" );
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document.write( " b =  .\r\n" );
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document.write( "You then substitute this expression for \"b\" into the first equation and get a unique solution for \"a\".\r\n" );
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document.write( "So, if m =/= 0, you always has the unique solution to the system of equations.\r\n" );
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document.write( "The only case, when you have infinitely many solutions for \"b\", is the case m= 0.\r\n" );
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document.write( "Then k must be -5 , according to equation (*).\r\n" );
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document.write( "In this way, you get the \r\n" );
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document.write( "ANSWER. For m= 0 and k= -5, the given system has infinitely many solutions.\r\n" );
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document.write( " Conversely, in order for the given system has infinitely many solutions, \"m\" must be zero and \"k\" must be -5.\r\n" );
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document.write( " Thus the necessary and sufficient condition for the given system to have infinitely many solutions\r\n" );
document.write( " is the condition m= 0, k= -5. \r\n" );
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document.write( "Solved.\r
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