document.write( "Question 1019348: For what value(s) of k, if any, will the system have no solution, a unique solution, and infinitely many solutions? (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
\n" ); document.write( "kx + 2y = 3
\n" ); document.write( "2x − 4y = −6
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Algebra.Com's Answer #635279 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "For what value(s) of k, if any, will the system have no solution, a unique solution, and infinitely many solutions? (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
\n" ); document.write( "kx + 2y = 3
\n" ); document.write( "2x - 4y = -6
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document.write( "Answer. If k = -1 then the system has infinitely many solutions.\r\n" );
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document.write( "        For any other value of k the system has a unique solution.\r\n" );
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document.write( "Solution. \r\n" );
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document.write( "If k=-1 then the system has the form\r\n" );
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document.write( "-x + 2y =  3,    (1)\r\n" );
document.write( "2x - 4y = -6.    (2)\r\n" );
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document.write( "Notice that the row of coefficients of the second equation is proportional \r\n" );
document.write( "to the row of coefficients of the first equation (with the proportionality coefficient -2). \r\n" );
document.write( "The right side terms are proportional with the same coefficient -2. \r\n" );
document.write( "It means that the system (1), (2) has infinitely many solutions.\r\n" );
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document.write( "Geometrically, equations (1) and (2) represent the same straight line in the coordinate plane.\r\n" );
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document.write( "For any other value of k the rows of coefficients are not proportional.\r\n" );
document.write( "So the system has a unique solution.\r\n" );
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\n" ); document.write( "\n" ); document.write( "See the lesson Geometric interpretation of the linear system of two equations in two unknowns in this site.
\n" ); document.write( "There you will find more details.\r
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