Solve (2-i)(x+yi)=(y-2xi)(3+i)+1+i/1-ifo

$$( 2 - i ) ( x + y i ) = ( y - 2 x i ) ( 3 + i ) + \frac { 1 + i } { 1 - i } fo$$

Solve for f

$$\left\{\begin{matrix}f=\frac{i\left(\left(-2+i\right)x+\left(2-i\right)y+\left(-6-2i\right)\xi \right)}{o}\text{, }&o\neq 0\\f\in \mathrm{C}\text{, }&x=y+\left(-2-2i\right)\xi \text{ and }o=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2-i by x+yi.

$$\left(2-i\right)x+\left(2-i\right)yi=\left(y-2\xi \right)\left(3+i\right)+\frac{1+i}{1-i}fo$$

Multiply 2-i and i to get 1+2i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(y-2\xi \right)\left(3+i\right)+\frac{1+i}{1-i}fo$$

Use the distributive property to multiply y-2\xi by 3+i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +\frac{1+i}{1-i}fo$$

Multiply both numerator and denominator of \frac{1+i}{1-i} by the complex conjugate of the denominator, 1+i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +\frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}fo$$

Do the multiplications in \frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +\frac{2i}{2}fo$$

Divide 2i by 2 to get i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +ifo$$

Swap sides so that all variable terms are on the left hand side.

$$\left(3+i\right)y+\left(-6-2i\right)\xi +ifo=\left(2-i\right)x+\left(1+2i\right)y$$

Subtract \left(3+i\right)y from both sides.

$$\left(-6-2i\right)\xi +ifo=\left(2-i\right)x+\left(1+2i\right)y-\left(3+i\right)y$$

Combine \left(1+2i\right)y and \left(-3-i\right)y to get \left(-2+i\right)y.

$$\left(-6-2i\right)\xi +ifo=\left(2-i\right)x+\left(-2+i\right)y$$

Subtract \left(-6-2i\right)\xi from both sides.

$$ifo=\left(2-i\right)x+\left(-2+i\right)y-\left(-6-2i\right)\xi$$

Multiply -1 and -6-2i to get 6+2i.

$$ifo=\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi$$

The equation is in standard form.

$$iof=\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi$$

Divide both sides by io.

$$\frac{iof}{io}=\frac{\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi }{io}$$

Dividing by io undoes the multiplication by io.

$$f=\frac{\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi }{io}$$

Divide \left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi by io.

$$f=-\frac{i\left(\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi \right)}{o}$$

Solve for o

$$\left\{\begin{matrix}o=\frac{i\left(\left(-2+i\right)x+\left(2-i\right)y+\left(-6-2i\right)\xi \right)}{f}\text{, }&f\neq 0\\o\in \mathrm{C}\text{, }&x=y+\left(-2-2i\right)\xi \text{ and }f=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2-i by x+yi.

$$\left(2-i\right)x+\left(2-i\right)yi=\left(y-2\xi \right)\left(3+i\right)+\frac{1+i}{1-i}fo$$

Multiply 2-i and i to get 1+2i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(y-2\xi \right)\left(3+i\right)+\frac{1+i}{1-i}fo$$

Use the distributive property to multiply y-2\xi by 3+i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +\frac{1+i}{1-i}fo$$

Multiply both numerator and denominator of \frac{1+i}{1-i} by the complex conjugate of the denominator, 1+i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +\frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}fo$$

Do the multiplications in \frac{\left(1+i\right)\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +\frac{2i}{2}fo$$

Divide 2i by 2 to get i.

$$\left(2-i\right)x+\left(1+2i\right)y=\left(3+i\right)y+\left(-6-2i\right)\xi +ifo$$

Swap sides so that all variable terms are on the left hand side.

$$\left(3+i\right)y+\left(-6-2i\right)\xi +ifo=\left(2-i\right)x+\left(1+2i\right)y$$

Subtract \left(3+i\right)y from both sides.

$$\left(-6-2i\right)\xi +ifo=\left(2-i\right)x+\left(1+2i\right)y-\left(3+i\right)y$$

Combine \left(1+2i\right)y and \left(-3-i\right)y to get \left(-2+i\right)y.

$$\left(-6-2i\right)\xi +ifo=\left(2-i\right)x+\left(-2+i\right)y$$

Subtract \left(-6-2i\right)\xi from both sides.

$$ifo=\left(2-i\right)x+\left(-2+i\right)y-\left(-6-2i\right)\xi$$

Multiply -1 and -6-2i to get 6+2i.

$$ifo=\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi$$

Divide both sides by if.

$$\frac{ifo}{if}=\frac{\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi }{if}$$

Dividing by if undoes the multiplication by if.

$$o=\frac{\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi }{if}$$

Divide \left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi by if.

$$o=-\frac{i\left(\left(2-i\right)x+\left(-2+i\right)y+\left(6+2i\right)\xi \right)}{f}$$