Explanation:
\({LHS}=\left[L^{3}\right]\)
\(RHS = {\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]^x}{\left[ {L{T^{ - 1}}} \right]^y}{\left[ {M{L^2}{T^{ - 1}}} \right]^z}\)
\( = \left[ {{M^{ - x + z}}{L^{3x + y + 2z}}{T^{ - 2x - y - z}}} \right]\)
According to homogeneity principle,
\({LHS}={RHS}\)
\(\because-x+z=0\)
\(\therefore z = x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
\(3 x+y+2 z=3\)
\({\rm{5 x + y = 3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{(2)}}\)
and \(-2 x-y-z=0\)
or \(-2 x-y-x=0\)
or \(-3 x-y=0\)
or \({\rm{3 x + y = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{(3)}}\)
From Eqs. (2) and (3), we get
\(2 x=3 \Rightarrow x=3 / 2\)
\(z=x=3 / 2 \Rightarrow 3 x+y=0\)
or \(\dfrac{9}{2}+y=0 \Rightarrow y=-\dfrac{9}{2}\)
\(\dfrac{x}{z}=\dfrac{3 / 2}{3 / 2}=1\).