document.write( "Question 1165251: Solve the linear system.\r
\n" ); document.write( "\n" ); document.write( " 3x + y + 6z = 0
\n" ); document.write( " −6x − 16z = 4
\n" ); document.write( " 8y − 17z = 17\r
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\n" ); document.write( "(i) find the LU-factorization of the coefficient matrix,
\n" ); document.write( "(ii) solve the lower triangular system Ly = b, and
\n" ); document.write( "(iii) solve the upper triangular system Ux = y. \n" ); document.write( "
Algebra.Com's Answer #789758 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "d=3, g=1, h=6, ad=-6,      ag+e=0,      ah+j=-16 \r\n" );
document.write( "             a(3)=-6  (-2)(1)+e=0  (-2)(6)+j=-16\r\n" );
document.write( "                a=-2       -2+e=0      -12+j=-16\r\n" );
document.write( "                              e=2          j=-4\r\n" );
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document.write( "  bd=0,        bg+ce=8,          bh+cj+f=-17\r\n" );
document.write( "b(3)=0   (0)(1)+c(2)=8  (0)(6)+(4)(-4)+f=-17\r\n" );
document.write( "   b=0            2c=8             -16+f=-17 \r\n" );
document.write( "                   c=4                 f=-1\r\n" );
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document.write( "[Yes, I know I could have gone to online solvers and\r\n" );
document.write( "not have to do that by hand.  But the whole reason for\r\n" );
document.write( "doing this problem is NOT to get the answer. Nobody in \r\n" );
document.write( "their right mind would solve a 3x3 system using this\r\n" );
document.write( "method!  But a computer would solve many 20+ x 20+ \r\n" );
document.write( "systems this way!  The reason for solving this is to \r\n" );
document.write( "learn what the computers do.]\r\n" );
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document.write( "Now we solve the lower triangular system, Lyk = bk\r\n" );
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document.write( "y1 = 0     -2y1+y2 = 4     0y1+4y2+y3 = 17\r\n" );
document.write( "          -2(0)+y2 = 4        4(4)+y3 = 17\r\n" );
document.write( "                y2 = 4          16+y3 = 17 \r\n" );
document.write( "                                   y3 = 1\r\n" );
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document.write( "Finally we solve U(x,y,z) = yk\r\n" );
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document.write( "     3x+y+6z = 0     2y-4z = 4     -z = 1\r\n" );
document.write( "                                    z = -1\r\n" );
document.write( "                  2y-4(-1) = 4\r\n" );
document.write( "                      2y+4 = 4\r\n" );
document.write( "                        2y = 0\r\n" );
document.write( "                         y = 0\r\n" );
document.write( "  3x+0+6(-1) = 0\r\n" );
document.write( "        3x-6 = 0\r\n" );
document.write( "          3x = 6\r\n" );
document.write( "           x = 2\r\n" );
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document.write( "So the solution to the system is (x,y,z) = (2,0,-1)\r\n" );
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document.write( "Edwin
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