Solve r=2a(2+cosθ)

$$r = 2 a ( 2 + \cos \theta )$$

Solve for a

$$a=\frac{r}{2\left(\cos(\theta )+2\right)}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2a by 2+\cos(\theta ).

$$r=4a+2a\cos(\theta )$$

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

$$4a+2a\cos(\theta )=r$$

Combine all terms containing a.

$$\left(4+2\cos(\theta )\right)a=r$$

The equation is in standard form.

$$\left(2\cos(\theta )+4\right)a=r$$

Divide both sides by 4+2\cos(\theta ).

$$\frac{\left(2\cos(\theta )+4\right)a}{2\cos(\theta )+4}=\frac{r}{2\cos(\theta )+4}$$

Dividing by 4+2\cos(\theta ) undoes the multiplication by 4+2\cos(\theta ).

$$a=\frac{r}{2\cos(\theta )+4}$$

Divide r by 4+2\cos(\theta ).

$$a=\frac{r}{2\left(\cos(\theta )+2\right)}$$

Solve for r

$$r=2a\left(\cos(\theta )+2\right)$$