Solve 4/x-3-3/x-1=5/x-1-4/x-4

$$\frac { 4 } { x - 3 } - \frac { 3 } { x - 1 } = \frac { 5 } { x - 1 } - \frac { 4 } { x - 4 }$$

Solve for x

$$x = \frac{17}{5} = 3\frac{2}{5} = 3.4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 1,3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right)\left(x-1\right), the least common multiple of x-3,x-1,x-4.

$$\left(x-4\right)\left(x-1\right)\times 4-\left(x-4\right)\left(x-3\right)\times 3=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Use the distributive property to multiply x-4 by x-1 and combine like terms.

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

Use the distributive property to multiply x^{2}-5x+4 by 4.

$$4x^{2}-20x+16-\left(x-4\right)\left(x-3\right)\times 3=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$4x^{2}-20x+16-\left(x^{2}-7x+12\right)\times 3=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Use the distributive property to multiply x^{2}-7x+12 by 3.

$$4x^{2}-20x+16-\left(3x^{2}-21x+36\right)=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

To find the opposite of 3x^{2}-21x+36, find the opposite of each term.

$$4x^{2}-20x+16-3x^{2}+21x-36=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

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

$$x^{2}-20x+16+21x-36=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Combine -20x and 21x to get x.

$$x^{2}+x+16-36=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Subtract 36 from 16 to get -20.

$$x^{2}+x-20=\left(x-4\right)\left(x-3\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x^{2}+x-20=\left(x^{2}-7x+12\right)\times 5-\left(x-3\right)\left(x-1\right)\times 4$$

Use the distributive property to multiply x^{2}-7x+12 by 5.

$$x^{2}+x-20=5x^{2}-35x+60-\left(x-3\right)\left(x-1\right)\times 4$$

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

$$x^{2}+x-20=5x^{2}-35x+60-\left(x^{2}-4x+3\right)\times 4$$

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

$$x^{2}+x-20=5x^{2}-35x+60-\left(4x^{2}-16x+12\right)$$

To find the opposite of 4x^{2}-16x+12, find the opposite of each term.

$$x^{2}+x-20=5x^{2}-35x+60-4x^{2}+16x-12$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}+x-20=x^{2}-35x+60+16x-12$$

Combine -35x and 16x to get -19x.

$$x^{2}+x-20=x^{2}-19x+60-12$$

Subtract 12 from 60 to get 48.

$$x^{2}+x-20=x^{2}-19x+48$$

Subtract x^{2} from both sides.

$$x^{2}+x-20-x^{2}=-19x+48$$

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

$$x-20=-19x+48$$

Add 19x to both sides.

$$x-20+19x=48$$

Combine x and 19x to get 20x.

$$20x-20=48$$

Add 20 to both sides.

$$20x=48+20$$

Add 48 and 20 to get 68.

$$20x=68$$

Divide both sides by 20.

$$x=\frac{68}{20}$$

Reduce the fraction \frac{68}{20} to lowest terms by extracting and canceling out 4.

$$x=\frac{17}{5}$$