Solve x-y/4+x=y/x-4

$$\frac { x - y } { 4 + x } = \frac { y } { x - 4 }$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{x-4}{2}\text{, }&x\neq -4\text{ and }x\neq 4\\y\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x+4\right), the least common multiple of 4+x,x-4.

$$\left(x-4\right)\left(x-y\right)=\left(x+4\right)y$$

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

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

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

$$x^{2}-xy-4x+4y=xy+4y$$

Subtract xy from both sides.

$$x^{2}-xy-4x+4y-xy=4y$$

Combine -xy and -xy to get -2xy.

$$x^{2}-2xy-4x+4y=4y$$

Subtract 4y from both sides.

$$x^{2}-2xy-4x+4y-4y=0$$

Combine 4y and -4y to get 0.

$$x^{2}-2xy-4x=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-2xy-4x=-x^{2}$$

Add 4x to both sides.

$$-2xy=-x^{2}+4x$$

The equation is in standard form.

$$\left(-2x\right)y=4x-x^{2}$$

Divide both sides by -2x.

$$\frac{\left(-2x\right)y}{-2x}=\frac{x\left(4-x\right)}{-2x}$$

Dividing by -2x undoes the multiplication by -2x.

$$y=\frac{x\left(4-x\right)}{-2x}$$

Divide x\left(4-x\right) by -2x.

$$y=\frac{x}{2}-2$$

Solve for y

$$\left\{\begin{matrix}y=\frac{x-4}{2}\text{, }&|x|\neq 4\\y\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-4\right)\left(x+4\right), the least common multiple of 4+x,x-4.

$$\left(x-4\right)\left(x-y\right)=\left(x+4\right)y$$

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

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

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

$$x^{2}-xy-4x+4y=xy+4y$$

Subtract xy from both sides.

$$x^{2}-xy-4x+4y-xy=4y$$

Combine -xy and -xy to get -2xy.

$$x^{2}-2xy-4x+4y=4y$$

Subtract 4y from both sides.

$$x^{2}-2xy-4x+4y-4y=0$$

Combine 4y and -4y to get 0.

$$x^{2}-2xy-4x=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-2xy-4x=-x^{2}$$

Add 4x to both sides.

$$-2xy=-x^{2}+4x$$

The equation is in standard form.

$$\left(-2x\right)y=4x-x^{2}$$

Divide both sides by -2x.

$$\frac{\left(-2x\right)y}{-2x}=\frac{x\left(4-x\right)}{-2x}$$

Dividing by -2x undoes the multiplication by -2x.

$$y=\frac{x\left(4-x\right)}{-2x}$$

Divide x\left(4-x\right) by -2x.

$$y=\frac{x}{2}-2$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=2\left(y+2\right)\text{, }&y\neq -4\text{ and }y\neq 0\end{matrix}\right.$$