Solve R l _ { k } = ∫ (from 0 to l) of w(w)x wrt x

$$R l _ { k } = \int _ { 0 } ^ { l } w ( w ) x d x$$

Solve for R

$$\left\{\begin{matrix}R=\frac{\left(lw\right)^{2}}{2l_{k}}\text{, }&l_{k}\neq 0\\R\in \mathrm{R}\text{, }&\left(w=0\text{ or }l=0\right)\text{ and }l_{k}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$Rl_{k}=\int _{0}^{l}w^{2}x\mathrm{d}x$$

The equation is in standard form.

$$l_{k}R=\frac{l^{2}w^{2}}{2}$$

Divide both sides by l_{k}.

$$\frac{l_{k}R}{l_{k}}=\frac{l^{2}w^{2}}{2l_{k}}$$

Dividing by l_{k} undoes the multiplication by l_{k}.

$$R=\frac{l^{2}w^{2}}{2l_{k}}$$

Solve for l

$$\left\{\begin{matrix}l=\frac{\sqrt{2Rl_{k}}}{w}\text{; }l=-\frac{\sqrt{2Rl_{k}}}{w}\text{, }&\left(R\leq 0\text{ or }l_{k}\geq 0\right)\text{ and }\left(l_{k}\leq 0\text{ or }R\geq 0\right)\text{ and }w\neq 0\\l\in \mathrm{R}\text{, }&\left(R=0\text{ or }l_{k}=0\right)\text{ and }w=0\end{matrix}\right.$$