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

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

Solve for x

$$x=\left(-4-i\right)y-2i$$

Steps for Solving Linear Equation

Use the distributive property to multiply y-1 by i.

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

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

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

Use the distributive property to multiply 2y+1 by i.

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

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

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

Subtract 2x from both sides.

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

Combine x and -2x to get -x.

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

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

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

Combine \left(3+2i\right)y and \left(1-i\right)y to get \left(4+i\right)y.

$$-x-i=\left(4+i\right)y+i$$

Add i to both sides.

$$-x=\left(4+i\right)y+i+i$$

Add i and i to get 2i.

$$-x=\left(4+i\right)y+2i$$

Divide both sides by -1.

$$\frac{-x}{-1}=\frac{\left(4+i\right)y+2i}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x=\frac{\left(4+i\right)y+2i}{-1}$$

Divide \left(4+i\right)y+2i by -1.

$$x=\left(-4-i\right)y-2i$$

Solve for y

$$y=\left(-\frac{4}{17}+\frac{1}{17}i\right)x+\left(-\frac{2}{17}-\frac{8}{17}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply y-1 by i.

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

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

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

Use the distributive property to multiply 2y+1 by i.

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

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

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

Subtract \left(3+2i\right)y from both sides.

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

Combine \left(-1+i\right)y and \left(-3-2i\right)y to get \left(-4-i\right)y.

$$x+\left(-4-i\right)y-i=2x+i$$

Subtract x from both sides.

$$\left(-4-i\right)y-i=2x+i-x$$

Combine 2x and -x to get x.

$$\left(-4-i\right)y-i=x+i$$

Add i to both sides.

$$\left(-4-i\right)y=x+i+i$$

Add i and i to get 2i.

$$\left(-4-i\right)y=x+2i$$

Divide both sides by -4-i.

$$\frac{\left(-4-i\right)y}{-4-i}=\frac{x+2i}{-4-i}$$

Dividing by -4-i undoes the multiplication by -4-i.

$$y=\frac{x+2i}{-4-i}$$

Divide x+2i by -4-i.

$$y=\left(-\frac{4}{17}+\frac{1}{17}i\right)x+\left(-\frac{2}{17}-\frac{8}{17}i\right)$$