Solve (2x-y)+(y-x)i=1+3i

$$( 2 x - y ) + ( y - x ) i = 1 + 3 i$$

Solve for x

$$x=\left(\frac{3}{5}-\frac{1}{5}i\right)y+\left(-\frac{1}{5}+\frac{7}{5}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply y-x by i.

$$2x-y+iy-ix=1+3i$$

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

$$2x+\left(-1+i\right)y-ix=1+3i$$

Combine 2x and -ix to get \left(2-i\right)x.

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

Subtract \left(-1+i\right)y from both sides.

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

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

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

The equation is in standard form.

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

Divide both sides by 2-i.

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

Dividing by 2-i undoes the multiplication by 2-i.

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

Divide 1+3i+\left(1-i\right)y by 2-i.

$$x=\left(\frac{3}{5}-\frac{1}{5}i\right)y+\left(-\frac{1}{5}+\frac{7}{5}i\right)$$

Solve for y

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

Steps for Solving Linear Equation

Use the distributive property to multiply y-x by i.

$$2x-y+iy-ix=1+3i$$

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

$$2x+\left(-1+i\right)y-ix=1+3i$$

Combine 2x and -ix to get \left(2-i\right)x.

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

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

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

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

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

The equation is in standard form.

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

Divide both sides by -1+i.

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

Dividing by -1+i undoes the multiplication by -1+i.

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

Divide 1+3i+\left(-2+i\right)x by -1+i.

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