a*x+b*y+z=1 x+a*b*y+z=b x+b*y+a*z=1

x + a*b*y + z = b

x + b*y + a*z = 1

1 – y – z

$$b_{1} = 1$$
=
$$1$$
=

1

$$a_{1} = 1$$
=
$$1$$
=

1

z

$$b_{2} = – frac{1}{2 y^{2}} left(y z – y + zright) + frac{1}{2 y^{2}} sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}$$
=
$$frac{1}{2 y^{2}} left(- y z + y – z + sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}right)$$
=

0.5*(y^2 + z^2 + 2*y*z^2 + 9*y^2*z^2 – 2*y*z – 2*z*y^2)^0.5/y^2 – 0.5*(z – y + y*z)/y^2

$$a_{2} = frac{1}{2 y z} left(- y z + y + z – sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}right)$$
=
$$frac{1}{2 y z} left(- y z + y + z – sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}right)$$
=

0.5*(y + z – (y^2 + z^2 + 2*y*z^2 + 9*y^2*z^2 – 2*y*z – 2*z*y^2)^0.5 – y*z)/(y*z)

z

$$b_{3} = – frac{1}{2 y^{2}} left(y z – y + zright) – frac{1}{2 y^{2}} sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}$$
=
$$frac{1}{2 y^{2}} left(- y z + y – z – sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}right)$$
=

-0.5*(y^2 + z^2 + 2*y*z^2 + 9*y^2*z^2 – 2*y*z – 2*z*y^2)^0.5/y^2 – 0.5*(z – y + y*z)/y^2

$$a_{3} = frac{1}{2 y z} left(- y z + y + z + sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}right)$$
=
$$frac{1}{2 y z} left(- y z + y + z + sqrt{9 y^{2} z^{2} – 2 y^{2} z + y^{2} + 2 y z^{2} – 2 y z + z^{2}}right)$$
=

0.5*(y + z + (y^2 + z^2 + 2*y*z^2 + 9*y^2*z^2 – 2*y*z – 2*z*y^2)^0.5 – y*z)/(y*z)