Solution (i) {x1+2x2−2x3=−12x1+3x2+x3=15x1+4x2−3x3=1\begin{cases} x_1+2x_2-2x_3=-1\\ 2x_1+3x_2+x_3=1\\ 5x_1+4x_2-3x_3=1 \end{cases}⎩⎨⎧x1+2x2−2x3=−12x1+3x2+x3=15x1+4x2−3x3=1 RREF gives: x1=1923, x2=−923, x3=1223\boxed{x_1=\frac{19}{23},\; x_2=-\frac{9}{23},\; x_3=\frac{12}{23}}x1=2319,x2=−239,x3=2312 (ii) {x+2y+z=22x+y+2z=32x+3y−z=7\begin{cases} x+2y+z=2\\ 2x+y+2z=3\\ 2x+3y-z=7 \end{cases}⎩⎨⎧x+2y+z=22x+y+2z=32x+3y−z=7 RREF gives: x=229, y=13, z=−109\boxed{x=\frac{22}{9},\; y=\frac{1}{3},\; z=-\frac{10}{9}}x=922,y=31,z=−910 (iii) {x1+4x2+x3=22x1+x2−2x3=93x1+x2−x3=12\begin{cases} x_1+4x_2+x_3=2\\ 2x_1+x_2-2x_3=9\\ 3x_1+x_2-x_3=12 \end{cases}⎩⎨⎧x1+4x2+x3=22x1+x2−2x3=93x1+x2−x3=12 RREF gives: x1=6116, x2=−14, x3=−1316\boxed{x_1=\frac{61}{16},\; x_2=-\frac{1}{4},\; x_3=-\frac{13}{16}}x1=1661,x2=−41,x3=−1613