document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120019>Question 1120019</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. If the system is dependent, express x, y, and z in terms of the parameter a.\r<BR>\n" );
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document.write( "3x+3y+12z=3<BR>\n" );
document.write( "x+y+4z=1<BR>\n" );
document.write( "2x+5y+20z=8<BR>\n" );
document.write( "-x+2y+8z=5\r<BR>\n" );
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document.write( "**NOTE: I've determined the determinate is zero and tried entering \"no solution\" but that is incorrect. I've also figured -1,2,0,0 </SPAN>\n" );
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document.write( "The system has only three variables; the answer must be an ordered set of three numbers, not four.<br><BR>\n" );
document.write( "Quick inspection shows that the first equation is three times the second.  That means the first equation is superfluous; it also means the determinant will be zero.<br><BR>\n" );
document.write( "You say you have \"figured -1,2,0,0\".  I will guess that you got that using matrices.  But what you show is only part of the story.<br><BR>\n" );
document.write( "As pointed out earlier, what you really found was -1,2,0, since there are only three variables.<br><BR>\n" );
document.write( "Using only the last three of the four equations, the matrix you end up with is<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "  1  0  0 -1\r\n" );
document.write( "  0  1  4  2\r\n" );
document.write( "  0  0  0  0</PRE><br><BR>\n" );
document.write( "The first row says<br><BR>\n" );
document.write( "x = -1<br><BR>\n" );
document.write( "The second row says<br><BR>\n" );
document.write( "y+4z = 2<br><BR>\n" );
document.write( "The third row says 0 = 0, which means the system is dependent, with an infinite family of solutions.<br><BR>\n" );
document.write( "So solve the equation from the second row for one of the variables and use that to define the set of solutions using the parameter a.<br><BR>\n" );
document.write( "y + 4z = 2<BR>\n" );
document.write( "y = 2-4z<br><BR>\n" );
document.write( "Then the solution set in parametric form is<br><BR>\n" );
document.write( "x = -1<BR>\n" );
document.write( "y = 2-4a<BR>\n" );
document.write( "z = a </SPAN>\n" );
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