Solve (x+2y-3)dy-(2x-y+1)dx=0

$$( x + 2 y - 3 ) d y - ( 2 x - y + 1 ) d x = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{ or }x=\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(xd+2yd-3d\right)y-\left(2x-y+1\right)dx=0$$

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

$$xdy+2dy^{2}-3dy-\left(2x-y+1\right)dx=0$$

Use the distributive property to multiply 2x-y+1 by d.

$$xdy+2dy^{2}-3dy-\left(2xd-yd+d\right)x=0$$

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

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

To find the opposite of 2dx^{2}-ydx+dx, find the opposite of each term.

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

Combine xdy and ydx to get 2xdy.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=-\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{ and }y\leq -\frac{\sqrt{11}}{5}+\frac{7}{10}\right)\text{ or }\left(x=-\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{ and }y\geq \frac{\sqrt{11}}{5}+\frac{7}{10}\right)\text{ or }\left(x=\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{ and }y\leq -\frac{\sqrt{11}}{5}+\frac{7}{10}\right)\text{ or }\left(x=\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{ and }y\geq \frac{\sqrt{11}}{5}+\frac{7}{10}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(xd+2yd-3d\right)y-\left(2x-y+1\right)dx=0$$

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

$$xdy+2dy^{2}-3dy-\left(2x-y+1\right)dx=0$$

Use the distributive property to multiply 2x-y+1 by d.

$$xdy+2dy^{2}-3dy-\left(2xd-yd+d\right)x=0$$

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

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

To find the opposite of 2dx^{2}-ydx+dx, find the opposite of each term.

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

Combine xdy and ydx to get 2xdy.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{; }x=\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{; }x=\frac{\sqrt{20y^{2}-28y+1}}{4}+\frac{y}{2}-\frac{1}{4}\text{, }&y\geq \frac{\sqrt{11}}{5}+\frac{7}{10}\text{ or }y\leq -\frac{\sqrt{11}}{5}+\frac{7}{10}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$