Solve r=xcosθ+ysinθ

$$r = x \cos \theta + y \sin \theta$$

Solve for x

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

Steps for Solving Linear Equation

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

$$x\cos(\theta )+y\sin(\theta )=r$$

Subtract y\sin(\theta ) from both sides.

$$x\cos(\theta )=r-y\sin(\theta )$$

The equation is in standard form.

$$\cos(\theta )x=-y\sin(\theta )+r$$

Divide both sides by \cos(\theta ).

$$\frac{\cos(\theta )x}{\cos(\theta )}=\frac{-y\sin(\theta )+r}{\cos(\theta )}$$

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

$$x=\frac{-y\sin(\theta )+r}{\cos(\theta )}$$

Solve for r

$$r=x\cos(\theta )+y\sin(\theta )$$