Solve 2x+y+x-(x+6y)dy=0

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

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{3x+y}{y\left(x+6y\right)}\text{, }&y\neq 0\text{ and }x\neq -6y\\d\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine 2x and x to get 3x.

$$3x+y-\left(x+6y\right)dy=0$$

Use the distributive property to multiply x+6y by d.

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

Use the distributive property to multiply xd+6yd by y.

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

To find the opposite of xdy+6dy^{2}, find the opposite of each term.

$$3x+y-xdy-6dy^{2}=0$$

Subtract 3x from both sides. Anything subtracted from zero gives its negation.

$$y-xdy-6dy^{2}=-3x$$

Subtract y from both sides.

$$-xdy-6dy^{2}=-3x-y$$

Combine all terms containing d.

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

Divide both sides by -yx-6y^{2}.

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

Dividing by -yx-6y^{2} undoes the multiplication by -yx-6y^{2}.

$$d=\frac{-3x-y}{-xy-6y^{2}}$$

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

$$d=\frac{3x+y}{y\left(x+6y\right)}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{3x+y}{y\left(x+6y\right)}\text{, }&y\neq 0\text{ and }x\neq -6y\\d\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine 2x and x to get 3x.

$$3x+y-\left(x+6y\right)dy=0$$

Use the distributive property to multiply x+6y by d.

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

Use the distributive property to multiply xd+6yd by y.

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

To find the opposite of xdy+6dy^{2}, find the opposite of each term.

$$3x+y-xdy-6dy^{2}=0$$

Subtract 3x from both sides. Anything subtracted from zero gives its negation.

$$y-xdy-6dy^{2}=-3x$$

Subtract y from both sides.

$$-xdy-6dy^{2}=-3x-y$$

Combine all terms containing d.

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

Divide both sides by -xy-6y^{2}.

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

Dividing by -xy-6y^{2} undoes the multiplication by -xy-6y^{2}.

$$d=\frac{-3x-y}{-xy-6y^{2}}$$

Divide -3x-y by -xy-6y^{2}.

$$d=\frac{3x+y}{y\left(x+6y\right)}$$

Solve for x

$$x=-\frac{y\left(1-6dy\right)}{3-dy}$$

$$d=0\text{ or }y\neq \frac{3}{d}$$

Steps for Solving Linear Equation

Combine 2x and x to get 3x.

$$3x+y-\left(x+6y\right)dy=0$$

Use the distributive property to multiply x+6y by d.

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

Use the distributive property to multiply xd+6yd by y.

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

To find the opposite of xdy+6dy^{2}, find the opposite of each term.

$$3x+y-xdy-6dy^{2}=0$$

Subtract y from both sides. Anything subtracted from zero gives its negation.

$$3x-xdy-6dy^{2}=-y$$

Add 6dy^{2} to both sides.

$$3x-xdy=-y+6dy^{2}$$

Combine all terms containing x.

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

The equation is in standard form.

$$\left(3-dy\right)x=6dy^{2}-y$$

Divide both sides by -dy+3.

$$\frac{\left(3-dy\right)x}{3-dy}=\frac{y\left(6dy-1\right)}{3-dy}$$

Dividing by -dy+3 undoes the multiplication by -dy+3.

$$x=\frac{y\left(6dy-1\right)}{3-dy}$$