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

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

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-2y+\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\text{ or }x=-2y-\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

Combine 2ydx and 2xdy to get 4ydx.

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

Combine all terms containing d.

$$\left(x^{2}+4yx-x+4y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}+4xy-x+4y^{2}\right)d=0$$

Divide 0 by x^{2}+4yx-x+4y^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=-2y+\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\text{ or }x=-2y-\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\right)\text{ and }y\leq \frac{1}{8}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

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

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

Combine 2ydx and 2xdy to get 4ydx.

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

Combine all terms containing d.

$$\left(x^{2}+4yx-x+4y^{2}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}+4xy-x+4y^{2}\right)d=0$$

Divide 0 by x^{2}+4yx-x+4y^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-2y+\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\text{; }x=-2y-\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-2y+\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\text{; }x=-2y-\frac{\sqrt{1-8y}}{2}+\frac{1}{2}\text{, }&y\leq \frac{1}{8}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$