Solve x^2/x^2-5x-14+3x/x+2+2x/x-7=5x+6/x+2+1

$$\frac { x ^ { 2 } } { x ^ { 2 } - 5 x - 14 } + \frac { 3 x } { x + 2 } + \frac { 2 x } { x - 7 } = \frac { 5 x + 6 } { x + 2 } + 1$$

Solve for x

$$x = -\frac{56}{17} = -3\frac{5}{17} \approx -3.294117647$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+2\right), the least common multiple of x^{2}-5x-14,x+2,x-7.

$$x^{2}+\left(x-7\right)\times 3x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by 3.

$$x^{2}+\left(3x-21\right)x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Use the distributive property to multiply 3x-21 by x.

$$x^{2}+3x^{2}-21x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

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

$$4x^{2}-21x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 2.

$$4x^{2}-21x+\left(2x+4\right)x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Use the distributive property to multiply 2x+4 by x.

$$4x^{2}-21x+2x^{2}+4x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Combine 4x^{2} and 2x^{2} to get 6x^{2}.

$$6x^{2}-21x+4x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Combine -21x and 4x to get -17x.

$$6x^{2}-17x=\left(x-7\right)\left(5x+6\right)+\left(x-7\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by 5x+6 and combine like terms.

$$6x^{2}-17x=5x^{2}-29x-42+\left(x-7\right)\left(x+2\right)$$

Use the distributive property to multiply x-7 by x+2 and combine like terms.

$$6x^{2}-17x=5x^{2}-29x-42+x^{2}-5x-14$$

Combine 5x^{2} and x^{2} to get 6x^{2}.

$$6x^{2}-17x=6x^{2}-29x-42-5x-14$$

Combine -29x and -5x to get -34x.

$$6x^{2}-17x=6x^{2}-34x-42-14$$

Subtract 14 from -42 to get -56.

$$6x^{2}-17x=6x^{2}-34x-56$$

Subtract 6x^{2} from both sides.

$$6x^{2}-17x-6x^{2}=-34x-56$$

Combine 6x^{2} and -6x^{2} to get 0.

$$-17x=-34x-56$$

Add 34x to both sides.

$$-17x+34x=-56$$

Combine -17x and 34x to get 17x.

$$17x=-56$$

Divide both sides by 17.

$$x=\frac{-56}{17}$$

Fraction \frac{-56}{17} can be rewritten as -\frac{56}{17} by extracting the negative sign.

$$x=-\frac{56}{17}$$