Solve x-4/x+7+1/3x-2=x-2/x+7

$$\frac { x - 4 } { x + 7 } + \frac { 1 } { 3 x - 2 } = \frac { x - 2 } { x + 7 }$$

Solve for x

$$x = \frac{11}{5} = 2\frac{1}{5} = 2.2$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -7,\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-2\right)\left(x+7\right), the least common multiple of x+7,3x-2.

$$\left(3x-2\right)\left(x-4\right)+x+7=\left(3x-2\right)\left(x-2\right)$$

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

$$3x^{2}-14x+8+x+7=\left(3x-2\right)\left(x-2\right)$$

Combine -14x and x to get -13x.

$$3x^{2}-13x+8+7=\left(3x-2\right)\left(x-2\right)$$

Add 8 and 7 to get 15.

$$3x^{2}-13x+15=\left(3x-2\right)\left(x-2\right)$$

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

$$3x^{2}-13x+15=3x^{2}-8x+4$$

Subtract 3x^{2} from both sides.

$$3x^{2}-13x+15-3x^{2}=-8x+4$$

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

$$-13x+15=-8x+4$$

Add 8x to both sides.

$$-13x+15+8x=4$$

Combine -13x and 8x to get -5x.

$$-5x+15=4$$

Subtract 15 from both sides.

$$-5x=4-15$$

Subtract 15 from 4 to get -11.

$$-5x=-11$$

Divide both sides by -5.

$$x=\frac{-11}{-5}$$

Fraction \frac{-11}{-5} can be simplified to \frac{11}{5} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{11}{5}$$