Solve 1/2-x+a=a+1/x

$$\frac { 1 } { 2 - x } + a = \frac { a + 1 } { x }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{2}{x-2}\text{, }&x\neq 2\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-2\right), the least common multiple of 2-x,x.

$$-x+x\left(x-2\right)a=\left(x-2\right)\left(a+1\right)$$

Use the distributive property to multiply x by x-2.

$$-x+\left(x^{2}-2x\right)a=\left(x-2\right)\left(a+1\right)$$

Use the distributive property to multiply x^{2}-2x by a.

$$-x+x^{2}a-2xa=\left(x-2\right)\left(a+1\right)$$

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

$$-x+x^{2}a-2xa=xa+x-2a-2$$

Subtract xa from both sides.

$$-x+x^{2}a-2xa-xa=x-2a-2$$

Combine -2xa and -xa to get -3xa.

$$-x+x^{2}a-3xa=x-2a-2$$

Add 2a to both sides.

$$-x+x^{2}a-3xa+2a=x-2$$

Add x to both sides.

$$x^{2}a-3xa+2a=x-2+x$$

Combine x and x to get 2x.

$$x^{2}a-3xa+2a=2x-2$$

Combine all terms containing a.

$$\left(x^{2}-3x+2\right)a=2x-2$$

Divide both sides by x^{2}-3x+2.

$$\frac{\left(x^{2}-3x+2\right)a}{x^{2}-3x+2}=\frac{2x-2}{x^{2}-3x+2}$$

Dividing by x^{2}-3x+2 undoes the multiplication by x^{2}-3x+2.

$$a=\frac{2x-2}{x^{2}-3x+2}$$

Divide -2+2x by x^{2}-3x+2.

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

Solve for a

$$\left\{\begin{matrix}a=\frac{2}{x-2}\text{, }&x\neq 2\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-2\right), the least common multiple of 2-x,x.

$$-x+x\left(x-2\right)a=\left(x-2\right)\left(a+1\right)$$

Use the distributive property to multiply x by x-2.

$$-x+\left(x^{2}-2x\right)a=\left(x-2\right)\left(a+1\right)$$

Use the distributive property to multiply x^{2}-2x by a.

$$-x+x^{2}a-2xa=\left(x-2\right)\left(a+1\right)$$

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

$$-x+x^{2}a-2xa=xa+x-2a-2$$

Subtract xa from both sides.

$$-x+x^{2}a-2xa-xa=x-2a-2$$

Combine -2xa and -xa to get -3xa.

$$-x+x^{2}a-3xa=x-2a-2$$

Add 2a to both sides.

$$-x+x^{2}a-3xa+2a=x-2$$

Add x to both sides.

$$x^{2}a-3xa+2a=x-2+x$$

Combine x and x to get 2x.

$$x^{2}a-3xa+2a=2x-2$$

Combine all terms containing a.

$$\left(x^{2}-3x+2\right)a=2x-2$$

Divide both sides by x^{2}-3x+2.

$$\frac{\left(x^{2}-3x+2\right)a}{x^{2}-3x+2}=\frac{2x-2}{x^{2}-3x+2}$$

Dividing by x^{2}-3x+2 undoes the multiplication by x^{2}-3x+2.

$$a=\frac{2x-2}{x^{2}-3x+2}$$

Divide -2+2x by x^{2}-3x+2.

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

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=2+\frac{2}{a}\text{, }&a\neq 0\text{ and }a\neq -1\end{matrix}\right.$$