How to solve this this equations? − 555 = 0.862 X + 0.138 Y −...

Hint:
Solve the first equation for $X$.
Substitute $X$ into the second equation and solve for $Y$.
Substitute $Y$ into the third equation and find $Z$.
Use this to find $Y$ and $X$ from the first two equations.
You should end up with:
$X=79.2468,Y=-140.527,Z=1750.49$
Note that there are many other ways to solve this, for example Gaussian Elimination or Cramer's Rule, etc.

The alternative to usual precise methods (Gauss–Jordan elimination used in most online calculators) is iterative methods. It may be Jacobi, or Gauss-Seidel (classical).