Solve x-2/x-1+5/3x-4=x+6/x-1

$$\frac { x - 2 } { x - 1 } + \frac { 5 } { 3 x - 4 } = \frac { x + 6 } { x - 1 }$$

Solve for x

$$x = \frac{27}{19} = 1\frac{8}{19} \approx 1.421052632$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply 3x-4 by x-2 and combine like terms.

$$3x^{2}-10x+8+\left(x-1\right)\times 5=\left(3x-4\right)\left(x+6\right)$$

Use the distributive property to multiply x-1 by 5.

$$3x^{2}-10x+8+5x-5=\left(3x-4\right)\left(x+6\right)$$

Combine -10x and 5x to get -5x.

$$3x^{2}-5x+8-5=\left(3x-4\right)\left(x+6\right)$$

Subtract 5 from 8 to get 3.

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

Use the distributive property to multiply 3x-4 by x+6 and combine like terms.

$$3x^{2}-5x+3=3x^{2}+14x-24$$

Subtract 3x^{2} from both sides.

$$3x^{2}-5x+3-3x^{2}=14x-24$$

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

$$-5x+3=14x-24$$

Subtract 14x from both sides.

$$-5x+3-14x=-24$$

Combine -5x and -14x to get -19x.

$$-19x+3=-24$$

Subtract 3 from both sides.

$$-19x=-24-3$$

Subtract 3 from -24 to get -27.

$$-19x=-27$$

Divide both sides by -19.

$$x=\frac{-27}{-19}$$

Fraction \frac{-27}{-19} can be simplified to \frac{27}{19} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{27}{19}$$