Solve 3/x-1+1/x-3=4/x-2

$$\frac { 3 } { x - 1 } + \frac { 1 } { x - 3 } = \frac { 4 } { x - 2 }$$

Solve for x

$$x=4$$

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-3,x-2.

$$\left(x-3\right)\left(x-2\right)\times 3+\left(x-2\right)\left(x-1\right)=\left(x-3\right)\left(x-1\right)\times 4$$

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

$$\left(x^{2}-5x+6\right)\times 3+\left(x-2\right)\left(x-1\right)=\left(x-3\right)\left(x-1\right)\times 4$$

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

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

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

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Combine -15x and -3x to get -18x.

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

Add 18 and 2 to get 20.

$$4x^{2}-18x+20=\left(x-3\right)\left(x-1\right)\times 4$$

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

$$4x^{2}-18x+20=\left(x^{2}-4x+3\right)\times 4$$

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

$$4x^{2}-18x+20=4x^{2}-16x+12$$

Subtract 4x^{2} from both sides.

$$4x^{2}-18x+20-4x^{2}=-16x+12$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-18x+20=-16x+12$$

Add 16x to both sides.

$$-18x+20+16x=12$$

Combine -18x and 16x to get -2x.

$$-2x+20=12$$

Subtract 20 from both sides.

$$-2x=12-20$$

Subtract 20 from 12 to get -8.

$$-2x=-8$$

Divide both sides by -2.

$$x=\frac{-8}{-2}$$

Divide -8 by -2 to get 4.

$$x=4$$