Find all complex solutions of z of the equation $2z+3\bar z=5$

Question

I am having trouble starting this problem, I'm suppose to solve for the variable $z$, where $z$ is a complex number ($z = x + yi$) and $\bar z$ is a complex conjugate ($\bar z = x - yi$) the question is...

"Find all complex solutions of $z$ of the equation $2 z+3{\bar z}=5$"

This is my worked solution so far;

$2z+3\bar z=5$

$2(x + yi) + 3(x - yi) = 5$

$2x + 2yi + 3x - 3yi = 5$

$5x - yi = 5$

At the last part, I'm not sure about how I solve for $x$ and $y$, and how I use the variables to solve for $z$.

Any help would be appreciated.

Answer 1

well I won't post a direct answer but take $w=2z+3\overline{z}$ as we know that $w$ is real we have the equation $ w = \overline{w} $ , and observe what I am suggesting you to ponder over !

Answer 2

That is simply $$ \eqalign{ & 2z + 3\bar z = 5\quad \Rightarrow \cr & \Rightarrow \quad 2\left( {x + iy} \right) + 3\left( {x - iy} \right) = 5\quad \Rightarrow \cr & \Rightarrow \quad 5x - iy = 5 + i0\quad \Rightarrow \cr & \Rightarrow \quad \left\{ \matrix{ x = 1 \hfill \cr y = 0 \hfill \cr} \right.\quad \Rightarrow \quad z = 1 \cr} $$

Answer 3

From the given, we have, $$2z+3\bar z=5,\>\>\>\>\>\>2\bar z+ 3z=5$$

Eliminate $\bar z$ in the system of equations above to obtain the solution $z=1$.