Solve (2x+y+1)dx-(4x+2y-3)dy=0

$$( 2 x + y + 1 ) d x - ( 4 x + 2 y - 3 ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=\frac{\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{ or }x=\frac{-\sqrt{25y^{2}-30y+1}+3y-1}{4}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2xd+yd+d\right)x-\left(4x+2y-3\right)dy=0$$

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

$$2dx^{2}+ydx+dx-\left(4x+2y-3\right)dy=0$$

Use the distributive property to multiply 4x+2y-3 by d.

$$2dx^{2}+ydx+dx-\left(4xd+2yd-3d\right)y=0$$

Use the distributive property to multiply 4xd+2yd-3d by y.

$$2dx^{2}+ydx+dx-\left(4xdy+2dy^{2}-3dy\right)=0$$

To find the opposite of 4xdy+2dy^{2}-3dy, find the opposite of each term.

$$2dx^{2}+ydx+dx-4xdy-2dy^{2}+3dy=0$$

Combine ydx and -4xdy to get -3ydx.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=\frac{\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{ and }y\leq \frac{3-2\sqrt{2}}{5}\right)\text{ or }\left(x=\frac{\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{ and }y\geq \frac{2\sqrt{2}+3}{5}\right)\text{ or }\left(x=\frac{-\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{ and }y\leq \frac{3-2\sqrt{2}}{5}\right)\text{ or }\left(x=\frac{-\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{ and }y\geq \frac{2\sqrt{2}+3}{5}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(2xd+yd+d\right)x-\left(4x+2y-3\right)dy=0$$

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

$$2dx^{2}+ydx+dx-\left(4x+2y-3\right)dy=0$$

Use the distributive property to multiply 4x+2y-3 by d.

$$2dx^{2}+ydx+dx-\left(4xd+2yd-3d\right)y=0$$

Use the distributive property to multiply 4xd+2yd-3d by y.

$$2dx^{2}+ydx+dx-\left(4xdy+2dy^{2}-3dy\right)=0$$

To find the opposite of 4xdy+2dy^{2}-3dy, find the opposite of each term.

$$2dx^{2}+ydx+dx-4xdy-2dy^{2}+3dy=0$$

Combine ydx and -4xdy to get -3ydx.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{; }x=\frac{-\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{; }x=\frac{-\sqrt{25y^{2}-30y+1}+3y-1}{4}\text{, }&y\geq \frac{2\sqrt{2}+3}{5}\text{ or }y\leq \frac{3-2\sqrt{2}}{5}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$