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

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

Solve for b

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

$$a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by a.

$$aa-ba=\left(a+1\right)x$$

Multiply a and a to get a^{2}.

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

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

$$a^{2}-ba=ax+x$$

Subtract a^{2} from both sides.

$$-ba=ax+x-a^{2}$$

The equation is in standard form.

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

Divide both sides by -a.

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

Dividing by -a undoes the multiplication by -a.

$$b=\frac{ax+x-a^{2}}{-a}$$

Divide ax+x-a^{2} by -a.

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{-\sqrt{x^{2}+2bx+4x+b^{2}}+b+x}{2}\text{, }&\left(arg(b)\geq \pi \text{ and }b\neq 0\right)\text{ or }x\neq 0\\a=\frac{\sqrt{x^{2}+2bx+4x+b^{2}}+b+x}{2}\text{, }&\left(arg(b)<\pi \text{ and }b\neq 0\right)\text{ or }x\neq 0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=\frac{-\sqrt{x^{2}+2bx+4x+b^{2}}+b+x}{2}\text{, }&\left(b\leq -x-2\sqrt{-x}\text{ or }b\geq -x+2\sqrt{-x}\text{ or }x\geq 0\right)\text{ and }\left(x\neq 0\text{ or }b\leq 0\right)\text{ and }\left(b<0\text{ or }x\neq 0\right)\text{ and }\left(x>0\text{ or }b\geq -x+2\sqrt{-x}\text{ or }b\leq -x-2\sqrt{-x}\text{ or }b<0\right)\\a=\frac{\sqrt{x^{2}+2bx+4x+b^{2}}+b+x}{2}\text{, }&\left(b\leq -x-2\sqrt{-x}\text{ or }b\geq -x+2\sqrt{-x}\text{ or }x\geq 0\right)\text{ and }\left(x\neq 0\text{ or }b\geq 0\right)\text{ and }\left(b>0\text{ or }x\neq 0\right)\text{ and }\left(x>0\text{ or }b\geq -x+2\sqrt{-x}\text{ or }b\leq -x-2\sqrt{-x}\text{ or }b>0\right)\end{matrix}\right.$$