Solve 5x/x^2+x-6+2x-5/x^2-x-12=7x-10/x^2-6x+8

$$\frac{ 5x }{ { x }^{ 2 } +x-6 } + \frac{ 2x-5 }{ { x }^{ 2 } -x-12 } = \frac{ 7x-10 }{ { x }^{ 2 } -6x+8 }$$

Solve for x

$$x=1$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}+x-6,x^{2}-x-12,x^{2}-6x+8.

$$\left(x-4\right)\times 5x+\left(x-2\right)\left(2x-5\right)=\left(x+3\right)\left(7x-10\right)$$

Use the distributive property to multiply x-4 by 5.

$$\left(5x-20\right)x+\left(x-2\right)\left(2x-5\right)=\left(x+3\right)\left(7x-10\right)$$

Use the distributive property to multiply 5x-20 by x.

$$5x^{2}-20x+\left(x-2\right)\left(2x-5\right)=\left(x+3\right)\left(7x-10\right)$$

Use the distributive property to multiply x-2 by 2x-5 and combine like terms.

$$5x^{2}-20x+2x^{2}-9x+10=\left(x+3\right)\left(7x-10\right)$$

Combine 5x^{2} and 2x^{2} to get 7x^{2}.

$$7x^{2}-20x-9x+10=\left(x+3\right)\left(7x-10\right)$$

Combine -20x and -9x to get -29x.

$$7x^{2}-29x+10=\left(x+3\right)\left(7x-10\right)$$

Use the distributive property to multiply x+3 by 7x-10 and combine like terms.

$$7x^{2}-29x+10=7x^{2}+11x-30$$

Subtract 7x^{2} from both sides.

$$7x^{2}-29x+10-7x^{2}=11x-30$$

Combine 7x^{2} and -7x^{2} to get 0.

$$-29x+10=11x-30$$

Subtract 11x from both sides.

$$-29x+10-11x=-30$$

Combine -29x and -11x to get -40x.

$$-40x+10=-30$$

Subtract 10 from both sides.

$$-40x=-30-10$$

Subtract 10 from -30 to get -40.

$$-40x=-40$$

Divide both sides by -40.

$$x=\frac{-40}{-40}$$

Divide -40 by -40 to get 1.

$$x=1$$