Solve ab/a-x=b+b/ax-1

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

Solve for b (complex solution)

$$\left\{\begin{matrix}b=0\text{, }&\left(a\neq 0\text{ and }x=0\right)\text{ or }\left(a\neq x\text{ and }a\neq \frac{1}{x}\text{ and }x\neq 0\right)\\b\in \mathrm{C}\text{, }&\left(a\neq -1\text{ and }x=-1\right)\text{ or }\left(a\neq 1\text{ and }x=1\right)\text{ or }\left(a=0\text{ and }x\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-a\right)\left(ax-1\right), the least common multiple of a-x,ax-1.

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

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

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

Use the distributive property to multiply a-xa^{2} by b.

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

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

$$ab-xa^{2}b=\left(ax^{2}-x-xa^{2}+a\right)b+\left(x-a\right)b$$

Use the distributive property to multiply ax^{2}-x-xa^{2}+a by b.

$$ab-xa^{2}b=ax^{2}b-xb-xa^{2}b+ab+\left(x-a\right)b$$

Use the distributive property to multiply x-a by b.

$$ab-xa^{2}b=ax^{2}b-xb-xa^{2}b+ab+xb-ab$$

Combine -xb and xb to get 0.

$$ab-xa^{2}b=ax^{2}b-xa^{2}b+ab-ab$$

Combine ab and -ab to get 0.

$$ab-xa^{2}b=ax^{2}b-xa^{2}b$$

Subtract ax^{2}b from both sides.

$$ab-xa^{2}b-ax^{2}b=-xa^{2}b$$

Add xa^{2}b to both sides.

$$ab-xa^{2}b-ax^{2}b+xa^{2}b=0$$

Combine -xa^{2}b and xa^{2}b to get 0.

$$ab-ax^{2}b=0$$

Combine all terms containing b.

$$\left(a-ax^{2}\right)b=0$$

Divide 0 by a-ax^{2}.

$$b=0$$

Solve for b

$$\left\{\begin{matrix}b=0\text{, }&\left(a\neq 0\text{ and }x=0\right)\text{ or }\left(a\neq x\text{ and }a\neq \frac{1}{x}\text{ and }x\neq 0\right)\\b\in \mathrm{R}\text{, }&\left(a\neq -1\text{ and }x=-1\right)\text{ or }\left(a\neq 1\text{ and }x=1\right)\text{ or }\left(a=0\text{ and }x\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-a\right)\left(ax-1\right), the least common multiple of a-x,ax-1.

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

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

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

Use the distributive property to multiply a-xa^{2} by b.

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

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

$$ab-xa^{2}b=\left(ax^{2}-x-xa^{2}+a\right)b+\left(x-a\right)b$$

Use the distributive property to multiply ax^{2}-x-xa^{2}+a by b.

$$ab-xa^{2}b=ax^{2}b-xb-xa^{2}b+ab+\left(x-a\right)b$$

Use the distributive property to multiply x-a by b.

$$ab-xa^{2}b=ax^{2}b-xb-xa^{2}b+ab+xb-ab$$

Combine -xb and xb to get 0.

$$ab-xa^{2}b=ax^{2}b-xa^{2}b+ab-ab$$

Combine ab and -ab to get 0.

$$ab-xa^{2}b=ax^{2}b-xa^{2}b$$

Subtract ax^{2}b from both sides.

$$ab-xa^{2}b-ax^{2}b=-xa^{2}b$$

Add xa^{2}b to both sides.

$$ab-xa^{2}b-ax^{2}b+xa^{2}b=0$$

Combine -xa^{2}b and xa^{2}b to get 0.

$$ab-ax^{2}b=0$$

Combine all terms containing b.

$$\left(a-ax^{2}\right)b=0$$

Divide 0 by a-ax^{2}.

$$b=0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=0\text{, }&x\neq 0\\a\in \mathrm{C}\setminus 0\text{, }x=0\text{ or }\left(b\neq 0\text{ and }x\neq -1\text{ and }x\neq 1\right),\frac{1}{x},x\text{, }&\left(x\neq 0\text{ and }b=0\right)\text{ or }x=-1\text{ or }x=1\\a\neq 0\text{, }&b=0\text{ and }x=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&x\neq 0\\a\in \mathrm{R}\setminus 0\text{, }x=0\text{ or }\left(b\neq 0\text{ and }|x|\neq 1\right),\frac{1}{x},x\text{, }&\left(x\neq 0\text{ and }b=0\right)\text{ or }|x|=1\\a\neq 0\text{, }&b=0\text{ and }x=0\end{matrix}\right.$$