Solve dy/y+1=dx/x-1

$$\frac { d y } { y + 1 } = \frac { d x } { x - 1 }$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=0\text{, }&y\neq -1\text{ and }x\neq 1\\d\in \mathrm{C}\text{, }&y=-x\text{ and }x\neq 1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(y+1\right), the least common multiple of y+1,x-1.

$$\left(x-1\right)dy=\left(y+1\right)dx$$

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

$$\left(xd-d\right)y=\left(y+1\right)dx$$

Use the distributive property to multiply xd-d by y.

$$xdy-dy=\left(y+1\right)dx$$

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

$$xdy-dy=\left(yd+d\right)x$$

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

$$xdy-dy=ydx+dx$$

Subtract ydx from both sides.

$$xdy-dy-ydx=dx$$

Combine xdy and -ydx to get 0.

$$-dy=dx$$

Subtract dx from both sides.

$$-dy-dx=0$$

Combine all terms containing d.

$$\left(-y-x\right)d=0$$

The equation is in standard form.

$$\left(-x-y\right)d=0$$

Divide 0 by -y-x.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq -1\text{ and }x\neq 1\\d\in \mathrm{R}\text{, }&y=-x\text{ and }x\neq 1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(y+1\right), the least common multiple of y+1,x-1.

$$\left(x-1\right)dy=\left(y+1\right)dx$$

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

$$\left(xd-d\right)y=\left(y+1\right)dx$$

Use the distributive property to multiply xd-d by y.

$$xdy-dy=\left(y+1\right)dx$$

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

$$xdy-dy=\left(yd+d\right)x$$

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

$$xdy-dy=ydx+dx$$

Subtract ydx from both sides.

$$xdy-dy-ydx=dx$$

Combine xdy and -ydx to get 0.

$$-dy=dx$$

Subtract dx from both sides.

$$-dy-dx=0$$

Combine all terms containing d.

$$\left(-y-x\right)d=0$$

The equation is in standard form.

$$\left(-x-y\right)d=0$$

Divide 0 by -y-x.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=-y\text{, }&y\neq -1\\x\neq 1\text{, }&d=0\text{ and }y\neq -1\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(y+1\right), the least common multiple of y+1,x-1.

$$\left(x-1\right)dy=\left(y+1\right)dx$$

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

$$\left(xd-d\right)y=\left(y+1\right)dx$$

Use the distributive property to multiply xd-d by y.

$$xdy-dy=\left(y+1\right)dx$$

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

$$xdy-dy=\left(yd+d\right)x$$

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

$$xdy-dy=ydx+dx$$

Subtract ydx from both sides.

$$xdy-dy-ydx=dx$$

Combine xdy and -ydx to get 0.

$$-dy=dx$$

Swap sides so that all variable terms are on the left hand side.

$$dx=-dy$$

Divide both sides by d.

$$\frac{dx}{d}=-\frac{dy}{d}$$

Dividing by d undoes the multiplication by d.

$$x=-\frac{dy}{d}$$

Divide -dy by d.

$$x=-y$$

Variable x cannot be equal to 1.

$$x=-y\text{, }x\neq 1$$