[latex]egin{cases} x+2y+3+4t=1 \ 2x+3y+z+2t=1 \ x+y+z-t=1 \ x-2z-6t=1 end{cases}[/latex] [latex]left[egin{array}{cccccc}1&2&3&4&|&1\2&3&1&2&|&1\1&1&1&-1&|&1\1&0&-2&-6&|&1end{array} ight] \ \ Downarrow w_2 o w_2-2w_1 \ Downarrow w_3 o w_3-w_1 \ Downarrow w_4 o w_4-w_1 \ \ left[egin{array}{cccccc}1&2&3&4&|&1\0&-1&-5&-6&|&-1\0&-1&-2&-5&|&0\0&-2&-5&-10&|&0end{array} ight] \ \ Downarrow w_3 o w_3-w_2 \ Downarrow w_4 o w_4-2w_2[/latex] [latex]left[egin{array}{cccccc}1&2&3&4&|&1\0&-1&-5&-6&|&-1\0&0&3&1&|&1\0&0&5&2&|&2end{array} ight] \ \ Downarrow w_4 o w_4-frac{5}{3}w_3 \ \ left[egin{array}{cccccc}1&2&3&4&|&1\0&-1&-5&-6&|&-1\0&0&3&1&|&1\0&0&0&frac{1}{3}&|&frac{1}{3}end{array} ight][/latex] [latex]frac{1}{3}t=frac{1}{3} \ t=1 \ \ 3z+t=1 \ 3z+1=1 \ 3z=0 \ z=0 \ \ -y-5z-6t=-1 \ -y-6=-1 \ -y=5 \ y=-5 \ \ x+2y+3z+4t=1 \ x-10+4=1 \ x-6=1 \ x=7[/latex] [latex]egin{cases} x=7 \ y=-5 \ z=0 \ t=1 end{cases}[/latex]
