Solve 2/x-2+a+1/a-2=0

$$\frac { 2 } { x - 2 } + \frac { a + 1 } { a - 2 } = 0$$

Solve for a

$$a=-1+\frac{6}{x}$$

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

Steps for Solving Linear Equation

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

$$\left(a-2\right)\times 2+\left(x-2\right)\left(a+1\right)=0$$

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

$$2a-4+\left(x-2\right)\left(a+1\right)=0$$

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

$$2a-4+xa+x-2a-2=0$$

Combine 2a and -2a to get 0.

$$-4+xa+x-2=0$$

Subtract 2 from -4 to get -6.

$$-6+xa+x=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$xa+x=6$$

Subtract x from both sides.

$$xa=6-x$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{6-x}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{6-x}{x}$$

Divide 6-x by x.

$$a=-1+\frac{6}{x}$$

Variable a cannot be equal to 2.

$$a=-1+\frac{6}{x}\text{, }a\neq 2$$

Solve for x

$$x=\frac{6}{a+1}$$

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

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 \left(a-2\right)\left(x-2\right), the least common multiple of x-2,a-2.

$$\left(a-2\right)\times 2+\left(x-2\right)\left(a+1\right)=0$$

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

$$2a-4+\left(x-2\right)\left(a+1\right)=0$$

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

$$2a-4+xa+x-2a-2=0$$

Combine 2a and -2a to get 0.

$$-4+xa+x-2=0$$

Subtract 2 from -4 to get -6.

$$-6+xa+x=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$xa+x=6$$

Combine all terms containing x.

$$\left(a+1\right)x=6$$

Divide both sides by 1+a.

$$\frac{\left(a+1\right)x}{a+1}=\frac{6}{a+1}$$

Dividing by 1+a undoes the multiplication by 1+a.

$$x=\frac{6}{a+1}$$

Variable x cannot be equal to 2.

$$x=\frac{6}{a+1}\text{, }x\neq 2$$