Solve 1/x-9=6/x^2+10x+x-5/x^2+10x

$$\frac { 1 } { x - 9 } = \frac { 6 } { x ^ { 2 } + 10 x } + \frac { x - 5 } { x ^ { 2 } + 10 x }$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -10,0,9 since division by zero is not defined. Multiply both sides of the equation by x\left(x-9\right)\left(x+10\right), the least common multiple of x-9,x^{2}+10x.

$$x\left(x+10\right)=\left(x-9\right)\times 6+\left(x-9\right)\left(x-5\right)$$

Use the distributive property to multiply x by x+10.

$$x^{2}+10x=\left(x-9\right)\times 6+\left(x-9\right)\left(x-5\right)$$

Use the distributive property to multiply x-9 by 6.

$$x^{2}+10x=6x-54+\left(x-9\right)\left(x-5\right)$$

Use the distributive property to multiply x-9 by x-5 and combine like terms.

$$x^{2}+10x=6x-54+x^{2}-14x+45$$

Combine 6x and -14x to get -8x.

$$x^{2}+10x=-8x-54+x^{2}+45$$

Add -54 and 45 to get -9.

$$x^{2}+10x=-8x-9+x^{2}$$

Add 8x to both sides.

$$x^{2}+10x+8x=-9+x^{2}$$

Combine 10x and 8x to get 18x.

$$x^{2}+18x=-9+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}+18x-x^{2}=-9$$

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

$$18x=-9$$

Divide both sides by 18.

$$x=\frac{-9}{18}$$

Reduce the fraction \frac{-9}{18} to lowest terms by extracting and canceling out 9.

$$x=-\frac{1}{2}$$