document.write( "Question 1110274: x - y + 3z = -8
\n" ); document.write( "2x + 3y - z = 5
\n" ); document.write( "3x + 2y + 2kz = -3k \r
\n" ); document.write( "\n" ); document.write( "For which values of k will the system have infinitely many solutions?
\n" ); document.write( "^ Does this mean how do I make the last equations equal to each other, so that there's an infinite number of solutions?
\n" ); document.write( "b) For which values of k will the system have exactly one solution?
\n" ); document.write( "This part I don't get. Do I try to get rid of k? Do I find a number for k that will give me one solution? And how do I find this number? I'm really confused! \n" ); document.write( "
Algebra.Com's Answer #725277 by ikleyn(52787)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
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" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "a) For which values of k will the system have infinitely many solutions? \r\n" );
document.write( "\r\n" );
document.write( "b) For which values of k will the system have exactly one solution? \r\n" );
document.write( "
~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solution\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "Add equations (1) and (2)  (both sides). You will get\r\n" );
document.write( "\r\n" );
document.write( "3x + 2y + 2z  = -3      (4)\r\n" );
document.write( "\r\n" );
document.write( "3x + 2y + 2kz = -3k     (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "I placed eq(3) of the original system below equation (4). \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now subtract eq(4) from eq(3). You will get\r\n" );
document.write( "\r\n" );
document.write( "2z*(k-1) = -3(k-1).     (5)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now from (5), it should be clear to you, that the value k = 1 is a SPECIAL value.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "a)  If k= 1, then both sides of (5) are equal each other and are equal to zero.\r\n" );
document.write( "\r\n" );
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" );
document.write( "\r\n" );
document.write( "    restrictions to the unknowns x, y and z, comparing with the system of two equations (1) and (2).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
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" );
document.write( "\r\n" );
document.write( "    This last system has, OBVIOUSLY, infinitely many solutions.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Thus the answer to question a) is k= 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "b)  If, on the contrary, k=/=1,  then from (5) you have the unique solution z= \"-3%2F2\" for z.\r\n" );
document.write( "\r\n" );
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" );
document.write( "\r\n" );
document.write( "     The 2x2 coefficient matrix of this system has a non-zero determinant.\r\n" );
document.write( "\r\n" );
document.write( "     It provides the unique solution in x and y for this reduced system.\r\n" );
document.write( "\r\n" );
document.write( "     Thus the answer to question b) is k=/= 1.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );