Solve (x-2y+1)dx+(4x-3y-6)dy=0

$$( x - 2 y + 1 ) d x + ( 4 x - 3 y - 6 ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{ or }x=-\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(xd-2yd+d\right)x+\left(4x-3y-6\right)dy=0$$

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

$$dx^{2}-2ydx+dx+\left(4x-3y-6\right)dy=0$$

Use the distributive property to multiply 4x-3y-6 by d.

$$dx^{2}-2ydx+dx+\left(4xd-3yd-6d\right)y=0$$

Use the distributive property to multiply 4xd-3yd-6d by y.

$$dx^{2}-2ydx+dx+4xdy-3dy^{2}-6dy=0$$

Combine -2ydx and 4xdy to get 2ydx.

$$dx^{2}+2ydx+dx-3dy^{2}-6dy=0$$

Combine all terms containing d.

$$\left(x^{2}+2yx+x-3y^{2}-6y\right)d=0$$

The equation is in standard form.

$$\left(x^{2}+2xy+x-3y^{2}-6y\right)d=0$$

Divide 0 by x^{2}+2yx+x-3y^{2}-6y.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(x=\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{ and }y\leq \frac{-3\sqrt{5}-7}{8}\right)\text{ or }\left(x=\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{ and }y\geq \frac{3\sqrt{5}-7}{8}\right)\text{ or }\left(x=-\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{ and }y\leq \frac{-3\sqrt{5}-7}{8}\right)\text{ or }\left(x=-\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{ and }y\geq \frac{3\sqrt{5}-7}{8}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(xd-2yd+d\right)x+\left(4x-3y-6\right)dy=0$$

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

$$dx^{2}-2ydx+dx+\left(4x-3y-6\right)dy=0$$

Use the distributive property to multiply 4x-3y-6 by d.

$$dx^{2}-2ydx+dx+\left(4xd-3yd-6d\right)y=0$$

Use the distributive property to multiply 4xd-3yd-6d by y.

$$dx^{2}-2ydx+dx+4xdy-3dy^{2}-6dy=0$$

Combine -2ydx and 4xdy to get 2ydx.

$$dx^{2}+2ydx+dx-3dy^{2}-6dy=0$$

Combine all terms containing d.

$$\left(x^{2}+2yx+x-3y^{2}-6y\right)d=0$$

The equation is in standard form.

$$\left(x^{2}+2xy+x-3y^{2}-6y\right)d=0$$

Divide 0 by x^{2}+2yx+x-3y^{2}-6y.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{; }x=-\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{; }x=-\frac{\sqrt{16y^{2}+28y+1}}{2}-y-\frac{1}{2}\text{, }&y\geq \frac{3\sqrt{5}-7}{8}\text{ or }y\leq \frac{-3\sqrt{5}-7}{8}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$