how do I find the unknown coefficients of coupled equations

Question

How do I find the unknown constants $x, y, z$ interms of a and b, using Mathematica.

\begin{equation} \begin{split} 1&=-x+y+z \\ 0&=y\frac{-4a}{b-\sqrt{b^2+8a^2}}+z\frac{-4a}{b+\sqrt{b^2+8a^2}}\\ 0&=x+y+z \end{split} \end{equation}

1 = -x+y+z
0 = y(-4 a/(b-Sqrt[b^2 + 8 a^2])) + z(-4 a/(b + Sqrt[b^2+8 a^2]))
0 = x+y+z

where $a,b$ are real constants

Answer 1

You can use LinearSolve. The matrix:

mat = {{-1, 1, 1},
   {0, -4 a/(b - Sqrt[b^2 + 8 a^2]), -4 a/(b + Sqrt[b^2 + 8 a^2])},
   {1, 1, 1}
   };

The vector:

v = {1, 0, 0};

Solving:

LinearSolve[mat, v]

yields:

{-(1/2), (-b + Sqrt[8 a^2 + b^2])/(4 Sqrt[8 a^2 + b^2]), (
 b + Sqrt[8 a^2 + b^2])/(4 Sqrt[8 a^2 + b^2])}

You could use Solve for the system of equations (note == instead of =).

Answer 2

Doesn't change much in this particular instance, but you can use Refine to incorporate assumptions on the parameters in your equations. Keep in mind that expressions in inequality assumptions are assumed to be real.

eqns = Refine[{1 == -x + y + z,
               0 == y (-4 a/(b - Sqrt[b^2 + 8 a^2])) + z (-4 a/(b + Sqrt[b^2 + 8 a^2])),
               0 == x + y + z}, 
               Assumptions -> Element[x, Reals] && Element[y, Reals]]
(* {1 == -x + y + z,
    0 == -((4 a y)/(b - Sqrt[8 a^2 + b^2])) - (4 a z)/(b + Sqrt[8 a^2 + b^2]),
    0 == x + y + z} *)

Solve[eqns, {x, y, z}]
    (* {{x -> -(1/2),
         y -> -((b - Sqrt[8 a^2 + b^2])/(4 Sqrt[8 a^2 + b^2])),
         z -> -((-b - Sqrt[8 a^2 + b^2])/(4 Sqrt[8 a^2 + b^2]))}} *)

Reduce[eqns, {x, y, z}]
    (* 8 a^2 + b^2 != 0 && a != 0 && x == -(1/2) && 
       y == (8 a^2 + b^2 - b Sqrt[8 a^2 + b^2])/(4 (8 a^2 + b^2)) && 
       z == (8 a^2 + b^2 + b Sqrt[8 a^2 + b^2])/(4 (8 a^2 + b^2)) *)