Question 256489
{{{system(10X + Y + Z = 12, 8X + 2Y +Z  = 11, 20X - 10Y - 2Z = 8)}}}
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Solve one equation for one letter,
Substitute that in both the other equations, and simplify them.

Pick the second equation to solve for Z

{{{8X + 2Y +Z  = 11}}}
{{{Z=11-8X-2Y}}}

Substitute {{{(11-8X-2Y)}}} for {{{Z}}} in the other two

{{{10X + Y + Z = 12}}}
{{{10X + Y + (11-8X-2Y) = 12}}}
{{{10X + Y + 11-8X-2Y = 12}}}
{{{2X - Y + 11 = 12}}}
{{{2X - Y = 1}}}


{{{20X - 10Y - 2Z = 8}}}
{{{20X - 10Y - 2(11-8X-2Y) = 8}}}
{{{20X - 10Y - 22+16X+4Y = 8}}}
{{{36X -6Y - 22 = 8}}}
{{{36X -6Y = 30}}}
Notice that you can divide that equation through by 6
{{{6X-Y=5}}}

So now you have this system with only 2 equations
and 2 unknowns:

{{{system(2X - Y = 1,6X-Y=5)}}}

Solve the first equation for Y:

{{{2X - Y = 1}}}
{{{-Y=1-2X}}}
{{{Y=-1+2X}}}

Substitute {{{(-1+2X)}}} for Y in
{{{6X-Y=5}}}
{{{6X-(-1+2X)=5}}}
{{{6X+1-2X=5}}}
{{{4X+1=5}}}
{{{4X=4}}}
{{{X=1}}}

Substitute 1 for X in

{{{Y=-1+2X}}}
{{{Y=-1+2(1)}}}
{{{Y=-1+2}}}
{{{Y=1}}}

Substitute 1 for  X and 1 for Y in 

{{{Z=11-8X-2Y}}}
{{{Z=11-8(1)-2(1)}}}
{{{Z=11-8-2}}}
{{{Z=1}}}

So (X,Y,Z) = (1,1,1)

Edwin</pre>