How to derive the Formula by Dimensional Analysis method?

Derive the Formula by Dimensional Analysis method Let a physical quantity $x$ depends on the other quantity $P, Q$ and $R$. Then $x \propto(\mathrm{P})^{\mathrm{a}}(\mathrm{Q})^{\mathrm{b}}(\mathrm{R})^{\mathrm{c}}$ $x=\mathrm{k}(\mathrm{P})^{a}(\mathrm{Q})^{b}(\mathrm{R})^{c}$ Now consider dimensional formula of each quantity in both side - $a x_{1}+b x_{2}+c x_{3}=x$ ...(2) $a y_{1}+b y_{2}+c y_{3}=y$ ...(3) $a z_{1}+b z_{2}+c z_{3}=z$ ...(4) After solving equation $(2),(3)$ and $(4)$ value of...