Solve (x+2y-3)dx-(2x+y-3)dy=0

$$( x + 2 y - 3 ) d x - ( 2 x + y - 3 ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=y\text{ or }x=3-y\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+2y-3 by d.

$$\left(xd+2yd-3d\right)x-\left(2x+y-3\right)dy=0$$

Use the distributive property to multiply xd+2yd-3d by x.

$$dx^{2}+2ydx-3dx-\left(2x+y-3\right)dy=0$$

Use the distributive property to multiply 2x+y-3 by d.

$$dx^{2}+2ydx-3dx-\left(2xd+yd-3d\right)y=0$$

Use the distributive property to multiply 2xd+yd-3d by y.

$$dx^{2}+2ydx-3dx-\left(2xdy+dy^{2}-3dy\right)=0$$

To find the opposite of 2xdy+dy^{2}-3dy, find the opposite of each term.

$$dx^{2}+2ydx-3dx-2xdy-dy^{2}+3dy=0$$

Combine 2ydx and -2xdy to get 0.

$$dx^{2}-3dx-dy^{2}+3dy=0$$

Combine all terms containing d.

$$\left(x^{2}-3x-y^{2}+3y\right)d=0$$

Divide 0 by x^{2}-3x-y^{2}+3y.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=y\text{ or }x=3-y\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+2y-3 by d.

$$\left(xd+2yd-3d\right)x-\left(2x+y-3\right)dy=0$$

Use the distributive property to multiply xd+2yd-3d by x.

$$dx^{2}+2ydx-3dx-\left(2x+y-3\right)dy=0$$

Use the distributive property to multiply 2x+y-3 by d.

$$dx^{2}+2ydx-3dx-\left(2xd+yd-3d\right)y=0$$

Use the distributive property to multiply 2xd+yd-3d by y.

$$dx^{2}+2ydx-3dx-\left(2xdy+dy^{2}-3dy\right)=0$$

To find the opposite of 2xdy+dy^{2}-3dy, find the opposite of each term.

$$dx^{2}+2ydx-3dx-2xdy-dy^{2}+3dy=0$$

Combine 2ydx and -2xdy to get 0.

$$dx^{2}-3dx-dy^{2}+3dy=0$$

Combine all terms containing d.

$$\left(x^{2}-3x-y^{2}+3y\right)d=0$$

Divide 0 by x^{2}-3x-y^{2}+3y.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=y\text{; }x=3-y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=y\text{; }x=3-y\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$