document.write( "Question 1080988: By solving these equation
\n" ); document.write( "Px+y+z=6
\n" ); document.write( "3x-y+11z=6
\n" ); document.write( "2x+y+4z=q
\n" ); document.write( " Find the condition of
\n" ); document.write( " (i) a unique solution
\n" ); document.write( " (ii) no solution
\n" ); document.write( " (iii) infinitely many solutions \n" ); document.write( "

Algebra.Com's Answer #695071 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "By solving these equation
\n" ); document.write( "Px+y+z=6
\n" ); document.write( "3x-y+11z=6
\n" ); document.write( "2x+y+4z=q\r
\n" ); document.write( "\n" ); document.write( "Find the condition of
\n" ); document.write( " (i) a unique solution
\n" ); document.write( " (ii) no solution
\n" ); document.write( " (iii) infinitely many solutions
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~\r
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document.write( "Px + y +   z = 6    (1)\r\n" );
document.write( "3x - y + 11z = 6    (2)\r\n" );
document.write( "2x + y +  4z = q    (3)\r\n" );
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document.write( "Let me exclude \"y\" from the system. In other words, I want to reduce the given 3x3-system to 2x2-system for unknowns \"x\" and \"z\".\r\n" );
document.write( "For it, I will add eqns (1) and (2) to get the equation (4)\r\n" );
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document.write( "(P+3)x + 12z = 12.      (4)\r\n" );
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document.write( "Also, add eqns (2) and (3) to get the equation (5)\r\n" );
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document.write( "5x + 15z = 6+q.         (5)\r\n" );
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document.write( "So, instead of (1),(2) and (3) I have now the system of 2 linear eqns in 2 unknowns\r\n" );
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document.write( "(P+3)x + 12z = 12.      (4)\r\n" );
document.write( "    5x + 15z = 6+q.     (5)\r\n" );
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document.write( "The determinant of the coefficient matrix M is\r\n" );
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document.write( "det(M) = (P+3)*15 - 5*12 = 15P + 45 - 60 = 15P - 15.\r\n" );
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document.write( "The determinant is zero, 15P - 15 = 0, if and only if P = 1.\r\n" );
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document.write( "Having this, we can make our FIRST conclusion:\r\n" );
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document.write( "     If P is different from 1, P =/= 1, then the system has a unique solution for any value of the parameter q.\r\n" );
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document.write( "Now consider the case P = 1.\r\n" );
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document.write( "In this case the system (4),(5) takes the form\r\n" );
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document.write( "4x + 12z = 12,      \r\n" );
document.write( "5x + 15z =  6 + q.   \r\n" );
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document.write( "or, simplifying,\r\n" );
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document.write( " x +  3z =  3,        (6)\r\n" );
document.write( "5x + 15z =  6 + q.    (7)\r\n" );
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document.write( "Express x from (6):  x = 3 - 3z,  and substitute it into (7). You will get\r\n" );
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document.write( "5*(3-3z) + 15z = 6 + q.\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "15 - 15z + 15z = 6 + q,\r\n" );
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document.write( "15 = 6 + q,\r\n" );
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document.write( "q = 9.\r\n" );
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document.write( "Having this, we can make two next conclusions.\r\n" );
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document.write( "    For P = 1,  the system (6),(7) has INFINITELY MANY solutions, if q = 9.\r\n" );
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document.write( "    For P = 1, the system (6),(7) has NO solutions if q =/= 9.\r\n" );
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document.write( "It gives the FINAL solution for the original system (1),(2),(3):\r\n" );
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document.write( "   It has a unique solution if P =/=1.\r\n" );
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document.write( "   If P = 1, the system (1)-(3) has INFINITELY MANY solutions at q = 9.\r\n" );
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document.write( "   If P = 1, the system (1)-(3) has NO              solutions at q =/= 9.\r\n" );
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