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

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

Solve for a

$$a=x-2$$

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

Solution Steps

Multiply both sides of the equation by \left(x-2\right)\left(x-1\right), the least common multiple of x-1,\left(x-1\right)\left(x-2\right).

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

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

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

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$x^{2}-2x+\left(x^{2}-3x+2\right)\left(-1\right)=a$$

Use the distributive property to multiply x^{2}-3x+2 by -1.

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

Combine x^{2} and -x^{2} to get 0.

$$-2x+3x-2=a$$

Combine -2x and 3x to get x.

$$x-2=a$$

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

$$a=x-2$$

Solve for x

$$x=a+2$$

$$a\neq -1\text{ and }a\neq 0$$

Solution Steps

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right), the least common multiple of x-1,\left(x-1\right)\left(x-2\right).

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

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

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

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$x^{2}-2x+\left(x^{2}-3x+2\right)\left(-1\right)=a$$

Use the distributive property to multiply x^{2}-3x+2 by -1.

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

Combine x^{2} and -x^{2} to get 0.

$$-2x+3x-2=a$$

Combine -2x and 3x to get x.

$$x-2=a$$

Add 2 to both sides.

$$x=a+2$$

Variable x cannot be equal to any of the values 1,2.

$$x=a+2\text{, }x\neq 1\text{ and }x\neq 2$$