Solve 3/x-1-2/x-2=1/x-3

$$\frac{ 3 }{ x-1 } - \frac{ 2 }{ x-2 } = \frac{ 1 }{ x-3 }$$

Solve for x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 1,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x-1\right), the least common multiple of x-1,x-2,x-3.

$$\left(x-3\right)\left(x-2\right)\times 3-\left(x-3\right)\left(x-1\right)\times 2=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-3 by x-2 and combine like terms.

$$\left(x^{2}-5x+6\right)\times 3-\left(x-3\right)\left(x-1\right)\times 2=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}-5x+6 by 3.

$$3x^{2}-15x+18-\left(x-3\right)\left(x-1\right)\times 2=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-3 by x-1 and combine like terms.

$$3x^{2}-15x+18-\left(x^{2}-4x+3\right)\times 2=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}-4x+3 by 2.

$$3x^{2}-15x+18-\left(2x^{2}-8x+6\right)=\left(x-2\right)\left(x-1\right)$$

To find the opposite of 2x^{2}-8x+6, find the opposite of each term.

$$3x^{2}-15x+18-2x^{2}+8x-6=\left(x-2\right)\left(x-1\right)$$

Combine 3x^{2} and -2x^{2} to get x^{2}.

$$x^{2}-15x+18+8x-6=\left(x-2\right)\left(x-1\right)$$

Combine -15x and 8x to get -7x.

$$x^{2}-7x+18-6=\left(x-2\right)\left(x-1\right)$$

Subtract 6 from 18 to get 12.

$$x^{2}-7x+12=\left(x-2\right)\left(x-1\right)$$

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

$$x^{2}-7x+12=x^{2}-3x+2$$

Subtract x^{2} from both sides.

$$x^{2}-7x+12-x^{2}=-3x+2$$

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

$$-7x+12=-3x+2$$

Add 3x to both sides.

$$-7x+12+3x=2$$

Combine -7x and 3x to get -4x.

$$-4x+12=2$$

Subtract 12 from both sides.

$$-4x=2-12$$

Subtract 12 from 2 to get -10.

$$-4x=-10$$

Divide both sides by -4.

$$x=\frac{-10}{-4}$$

Reduce the fraction \frac{-10}{-4} to lowest terms by extracting and canceling out -2.

$$x=\frac{5}{2}$$