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

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

Solve for a

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

$$x\neq 0\text{ and }x\neq 2$$

Steps for Solving Linear Equation

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

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

Combine x and x to get 2x.

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

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

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

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

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

To find the opposite of x^{2}a-2xa, find the opposite of each term.

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

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

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

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

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

Solve for x

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