Solve x-2/x-4=x+4/x-2

$$\frac { x - 2 } { x - 4 } = \frac { x + 4 } { x - 2 }$$

Solve for x

$$x=5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-2\right), the least common multiple of x-4,x-2.

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

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

$$\left(x-2\right)^{2}=\left(x-4\right)\left(x+4\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4=\left(x-4\right)\left(x+4\right)$$

Consider \left(x-4\right)\left(x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$x^{2}-4x+4=x^{2}-16$$

Subtract x^{2} from both sides.

$$x^{2}-4x+4-x^{2}=-16$$

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

$$-4x+4=-16$$

Subtract 4 from both sides.

$$-4x=-16-4$$

Subtract 4 from -16 to get -20.

$$-4x=-20$$

Divide both sides by -4.

$$x=\frac{-20}{-4}$$

Divide -20 by -4 to get 5.

$$x=5$$