Solve r=a-bcosθ

$$r = a - b \cos \theta$$

Solve for b

$$\left\{\begin{matrix}b=\frac{a-r}{\cos(\theta )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\\b\in \mathrm{R}\text{, }&r=a\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a-b\cos(\theta )=r$$

Subtract a from both sides.

$$-b\cos(\theta )=r-a$$

The equation is in standard form.

$$\left(-\cos(\theta )\right)b=r-a$$

Divide both sides by -\cos(\theta ).

$$\frac{\left(-\cos(\theta )\right)b}{-\cos(\theta )}=\frac{r-a}{-\cos(\theta )}$$

Dividing by -\cos(\theta ) undoes the multiplication by -\cos(\theta ).

$$b=\frac{r-a}{-\cos(\theta )}$$

Divide r-a by -\cos(\theta ).

$$b=\frac{a-r}{\cos(\theta )}$$

Solve for a

$$a=b\cos(\theta )+r$$