Solve x+10/x+100/10x-x^2=x+20/x-10

$$\frac { x + 10 } { x } + \frac { 100 } { 10 x - x ^ { 2 } } = \frac { x + 20 } { x - 10 }$$

Solve for x

$$x=-10$$

Steps for Solving Linear Equation

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

$$\left(x-10\right)\left(x+10\right)-100=x\left(x+20\right)$$

Consider \left(x-10\right)\left(x+10\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 10.

$$x^{2}-100-100=x\left(x+20\right)$$

Subtract 100 from -100 to get -200.

$$x^{2}-200=x\left(x+20\right)$$

Use the distributive property to multiply x by x+20.

$$x^{2}-200=x^{2}+20x$$

Subtract x^{2} from both sides.

$$x^{2}-200-x^{2}=20x$$

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

$$-200=20x$$

Swap sides so that all variable terms are on the left hand side.

$$20x=-200$$

Divide both sides by 20.

$$x=\frac{-200}{20}$$

Divide -200 by 20 to get -10.

$$x=-10$$