Solve (a)x/1+i+y/1-2i=1

$$( a ) \frac { x } { 1 + i } + \frac { y } { 1 - 2 i } = 1$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{\left(-1+3i\right)y+\left(-5-5i\right)}{5x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&y=1-2i\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{y}{1-2i} from both sides.

$$a\times \frac{x}{1+i}=1-\frac{y}{1-2i}$$

The equation is in standard form.

$$\left(\frac{1}{2}-\frac{1}{2}i\right)xa=\left(-\frac{1}{5}-\frac{2}{5}i\right)y+1$$

Divide both sides by \left(\frac{1}{2}-\frac{1}{2}i\right)x.

$$\frac{\left(\frac{1}{2}-\frac{1}{2}i\right)xa}{\left(\frac{1}{2}-\frac{1}{2}i\right)x}=\frac{\left(-\frac{1}{5}-\frac{2}{5}i\right)y+1}{\left(\frac{1}{2}-\frac{1}{2}i\right)x}$$

Dividing by \left(\frac{1}{2}-\frac{1}{2}i\right)x undoes the multiplication by \left(\frac{1}{2}-\frac{1}{2}i\right)x.

$$a=\frac{\left(-\frac{1}{5}-\frac{2}{5}i\right)y+1}{\left(\frac{1}{2}-\frac{1}{2}i\right)x}$$

Divide 1+\left(-\frac{1}{5}-\frac{2}{5}i\right)y by \left(\frac{1}{2}-\frac{1}{2}i\right)x.

$$a=\frac{\left(\frac{1}{5}+\frac{1}{5}i\right)\left(\left(-1-2i\right)y+5\right)}{x}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\left(-1+3i\right)y+\left(-5-5i\right)}{5a}\text{, }&a\neq 0\\x\in \mathrm{C}\text{, }&y=1-2i\text{ and }a=0\end{matrix}\right.$$