Solve 2/w-3+1/w-1=5/w-1-2/w-2

$$\frac { 2 } { w - 3 } + \frac { 1 } { w - 1 } = \frac { 5 } { w - 1 } - \frac { 2 } { w - 2 }$$

Solve for w

$$w = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps for Solving Linear Equation

Variable w 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(w-3\right)\left(w-2\right)\left(w-1\right), the least common multiple of w-3,w-1,w-2.

$$\left(w-2\right)\left(w-1\right)\times 2+\left(w-3\right)\left(w-2\right)=\left(w-3\right)\left(w-2\right)\times 5-\left(w-3\right)\left(w-1\right)\times 2$$

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

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

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

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

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

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

Combine 2w^{2} and w^{2} to get 3w^{2}.

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

Combine -6w and -5w to get -11w.

$$3w^{2}-11w+4+6=\left(w-3\right)\left(w-2\right)\times 5-\left(w-3\right)\left(w-1\right)\times 2$$

Add 4 and 6 to get 10.

$$3w^{2}-11w+10=\left(w-3\right)\left(w-2\right)\times 5-\left(w-3\right)\left(w-1\right)\times 2$$

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

$$3w^{2}-11w+10=\left(w^{2}-5w+6\right)\times 5-\left(w-3\right)\left(w-1\right)\times 2$$

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

$$3w^{2}-11w+10=5w^{2}-25w+30-\left(w-3\right)\left(w-1\right)\times 2$$

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

$$3w^{2}-11w+10=5w^{2}-25w+30-\left(w^{2}-4w+3\right)\times 2$$

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

$$3w^{2}-11w+10=5w^{2}-25w+30-\left(2w^{2}-8w+6\right)$$

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

$$3w^{2}-11w+10=5w^{2}-25w+30-2w^{2}+8w-6$$

Combine 5w^{2} and -2w^{2} to get 3w^{2}.

$$3w^{2}-11w+10=3w^{2}-25w+30+8w-6$$

Combine -25w and 8w to get -17w.

$$3w^{2}-11w+10=3w^{2}-17w+30-6$$

Subtract 6 from 30 to get 24.

$$3w^{2}-11w+10=3w^{2}-17w+24$$

Subtract 3w^{2} from both sides.

$$3w^{2}-11w+10-3w^{2}=-17w+24$$

Combine 3w^{2} and -3w^{2} to get 0.

$$-11w+10=-17w+24$$

Add 17w to both sides.

$$-11w+10+17w=24$$

Combine -11w and 17w to get 6w.

$$6w+10=24$$

Subtract 10 from both sides.

$$6w=24-10$$

Subtract 10 from 24 to get 14.

$$6w=14$$

Divide both sides by 6.

$$w=\frac{14}{6}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

$$w=\frac{7}{3}$$