Solve a+1/b=a-1/b+b+1/a

$$\frac { a + 1 } { b } = \frac { a - 1 } { b } + \frac { b + 1 } { a }$$

Solve for a

$$a=\frac{b\left(b+1\right)}{2}$$

$$b\neq -1\text{ and }b\neq 0$$

Steps for Solving Linear Equation

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by ab, the least common multiple of b,a.

$$a\left(a+1\right)=a\left(a-1\right)+b\left(b+1\right)$$

Use the distributive property to multiply a by a+1.

$$a^{2}+a=a\left(a-1\right)+b\left(b+1\right)$$

Use the distributive property to multiply a by a-1.

$$a^{2}+a=a^{2}-a+b\left(b+1\right)$$

Use the distributive property to multiply b by b+1.

$$a^{2}+a=a^{2}-a+b^{2}+b$$

Subtract a^{2} from both sides.

$$a^{2}+a-a^{2}=-a+b^{2}+b$$

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

$$a=-a+b^{2}+b$$

Add a to both sides.

$$a+a=b^{2}+b$$

Combine a and a to get 2a.

$$2a=b^{2}+b$$

Divide both sides by 2.

$$\frac{2a}{2}=\frac{b\left(b+1\right)}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a=\frac{b\left(b+1\right)}{2}$$

Variable a cannot be equal to 0.

$$a=\frac{b\left(b+1\right)}{2}\text{, }a\neq 0$$

Solve for b

$$b=\frac{-\sqrt{8a+1}-1}{2}$$

$$b=\frac{\sqrt{8a+1}-1}{2}\text{, }a\neq 0\text{ and }a\geq -\frac{1}{8}$$