How to solve a complex system of equations with a conjugate

The system:

$$(10-i)z-(2-i)\bar{w}=26+8i$$ $$(3+2i)\bar{z}+(1-2i)w=8-4i$$

$$w$$ and $$z$$ are complex numbers.

So for this kind of problems I would usually put it in matrix form and solve from there. But here I cannot do that since some numbers are in conjugate form.

I tried getting from the conjugate number to the normal complex number by $$z=\bar{z}+\frac{z}{\bar{z}}$$ But it seems that this way I will get nowhere. How should I tackle this problem?

• Write $z=x+iy$ etc. and solve with real variables. Jan 5, 2020 at 19:47
• As said by Arthur, take the conjugate of one of the equations and solve. If this doesn't work (for another system), you still have the recourse of expanding the real/imaginary parts and solving a $4\times4$ system.
– user65203
Jan 5, 2020 at 20:05

Conjugate the entire second equation and you now have two equations in the unknowns $$z$$ and $$\bar w$$, that you can solve the usual way.

Here is the system of 4 equations, which Yves Daoust has mentioned in his comment. This system has no imaginary numbers, but has the same collection of solutions as your system with complex numbers. This system can be obtained by substituting $$z=x+iy \space$$ and $$\space w=a+ib \space$$ and then separating the real and imaginary parts.

$$\begin{matrix} 10x & + & y & - & 2a & + & b & = & \space\space 26 \\ 3x & + & 2y & + & a & + &2 b & = & \space\space\space 8 \\ -x & + & 10y & + & a & + & 2b & = & \space\space\space 8 \\ 2x & - & 3y & - & 2a & + & b & = & -4 \\ \end{matrix}$$

Here is the matrix form of the above system of equations.

The solution can be obtained using Cramers's rule, Gaussian elimination, or any other method. The solutions are, $$x=3, y=1.5, a=1.4, \space\rm{and}\space \it{b} \rm{=-2.7}.$$

Therefore, $$z=3+1.5i \space\space\rm{and}\space \space \it{w}\rm{=1.4-2.7}\it{i}.$$

This method works for systems with and without complex conjugate.