Question 831765
Let {{{u=1/x}}},{{{v=1/y}}},{{{w=1/z}}}
.
.
.
1.{{{u+2v-4w=1}}}
2.{{{2u+3v+8w=0}}}
3.{{{-u+9v+10w=5}}}
Use Gaussian elimination.
Multiply eq. 1 by 2 and subtract from eq. 2.
{{{2u+4v-8w-2u-3v-8w=2}}}
4.{{{v-16w=2}}}
Add eq. 1 and eq. 3,
{{{u+2v-4w-u+9v+10w=1+5}}}
5.{{{11v+6w=6}}}
Now multiply eq. 4 by 11 and subtract from eq. 5,
{{{11v-176w-11v-6w=22-6}}}
{{{-182w=16}}}
{{{w=-16/182}}}
{{{w=-8/91}}}
Now back substitute,
{{{v-16w=2}}}
{{{v-16(-8/91)=2}}}
{{{v+128/91=182/91}}}
{{{v=54/91}}}
And finally,
{{{u+2v-4w=1}}}
{{{u+2(54/91)-4(-8/91)=1}}}
{{{u+108/91+32/91=91/91}}}
{{{u+140/91=91/91}}}
{{{u=-49/91}}}
{{{u=-7/13}}}
Last step,
{{{u=1/x=-7/13}}}
{{{highlight(x=-13/7)}}}
{{{v=1/y=54/91}}}
{{{highlight(y=91/54)}}}
{{{w=1/z=-8/91}}}
{{{highlight(z=-91/8)}}}