Solve 1/2-x-1-ax/x-2=2

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

Solve for a

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x-2, the least common multiple of 2-x,x-2.

$$-1-\left(1-ax\right)=2\left(x-2\right)$$

To find the opposite of 1-ax, find the opposite of each term.

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

Subtract 1 from -1 to get -2.

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

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

$$-2+ax=2x-4$$

Add 2 to both sides.

$$ax=2x-4+2$$

Add -4 and 2 to get -2.

$$ax=2x-2$$

The equation is in standard form.

$$xa=2x-2$$

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Divide -2+2x by x.

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

Solve for x

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

$$a\neq 1\text{ and }a\neq 2$$

Steps for Solving Linear Equation

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2, the least common multiple of 2-x,x-2.

$$-1-\left(1-ax\right)=2\left(x-2\right)$$

To find the opposite of 1-ax, find the opposite of each term.

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

Subtract 1 from -1 to get -2.

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

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

$$-2+ax=2x-4$$

Subtract 2x from both sides.

$$-2+ax-2x=-4$$

Add 2 to both sides.

$$ax-2x=-4+2$$

Add -4 and 2 to get -2.

$$ax-2x=-2$$

Combine all terms containing x.

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

Divide both sides by a-2.

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

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

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

Variable x cannot be equal to 2.

$$x=-\frac{2}{a-2}\text{, }x\neq 2$$