document.write( "Question 1110516: Let K be a real number. Consider the following system of equations:\r
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Algebra.Com's Answer #725509 by ikleyn(52781)\"\" \"About 
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document.write( " x -  y + 3z =  -8      (1)\r\n" );
document.write( "2x + 3y -  z =   5      (2)\r\n" );
document.write( "3x + 2y + 2kz = -3k     (3)\r\n" );
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document.write( "A) for which values of K will the system have infinitely many solutions?\r\n" );
document.write( "B) For which values of K will the system have exactly one solution?\r\n" );
document.write( "C) For which values of K will the system have no solution?
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document.write( "Add equations (1) and (2)  (both sides). You will get\r\n" );
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document.write( "3x + 2y + 2z  = -3      (4)\r\n" );
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document.write( "3x + 2y + 2kz = -3k     (3)\r\n" );
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document.write( "I placed eq(3) of the original system below equation (4). \r\n" );
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document.write( "Now subtract eq(4) from eq(3). You will get\r\n" );
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document.write( "2z*(k-1) = -3(k-1).     (5)\r\n" );
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document.write( "Now from (5), it should be clear to you, that the value k = 1 is a SPECIAL value.\r\n" );
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document.write( "a)  If k= 1, then both sides of (5) are equal each other and are equal to zero.\r\n" );
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document.write( "    At k= 1, equation (3) is simply the sum of equations (1) and (3), and, therefore, does not bring any new information or any new \r\n" );
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document.write( "    restrictions to the unknowns x, y and z, comparing with the system of two equations (1) and (2).\r\n" );
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document.write( "    In other words, at k= 1 the system (1),(2),(3) is EQUIVALENT to the system of two equations (1) and (2).\r\n" );
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document.write( "    This last system has, OBVIOUSLY, infinitely many solutions.\r\n" );
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document.write( "    Thus the answer to question a) is k= 1.\r\n" );
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document.write( "b)  If, on the contrary, k=/=1,  then from (5) you have the unique solution z= \"-3%2F2\" for z.\r\n" );
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document.write( "    You can substitute it into equations (1) and (2), and then you will get the system of two equations in two unknowns x and y.\r\n" );
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document.write( "     The 2x2 coefficient matrix of this system has a non-zero determinant.\r\n" );
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document.write( "     It provides the unique solution in x and y for this reduced system.\r\n" );
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document.write( "     Thus the answer to question b) is k=/= 1.\r\n" );
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document.write( "c)   From the solution above we got that\r\n" );
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document.write( "         - at k = 1 the system has INFINITELY MANY solution;\r\n" );
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document.write( "         - at k =/= 1 the system has a unique solution.\r\n" );
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document.write( "     It exhausts all possible options for k; hence, the set of those {k} the system has no solution is EMPTY. \r\n" );
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