Drive $,x_A=\frac{\frac{{}_A}{M_A}}{\frac{{}_A}{M_A}+\frac{{}_B}{M_B}}$ from ${}_A=\frac{x_A{M}_A}{x_A{M}_A+x_B{M}_B}$

Concentrations in a binary mixture of A and $B$

$=$ total mass density $of$ the mixture

${}_A=$ mass density $of$ species $A$

${}_B=$ mass density $of$ species $B$

${}_A=$ mass fraction $of$ species $A=\frac{{}_A}{}$

${}_B=$ mass fraction $of$ species $B=\frac{{}_B}{}$

$={}_A+{}_B$

$1={}_A+{}_B$

Mass concentrat

${}_A=$ mass density of species $A$

${}_B=$ mass density of species $B$

${}_A=$ mass fraction of species $A=\frac{{}_A}{}$

${}_B=$ mass fraction of species $B=\frac{{}_B}{}$

$={}_A+{}_B$

Mole concentrations

Liquid or solid mixture

$c=$ molar density $of$ mixture $=\frac{n}{V}$

$c_A=$ molar density $of$ species $A=\frac{n_A}{V}$

$c_B=$ molar density $of$ species $B=\frac{n_B}{V}$

$x_A=$ mole fraction $of$ species $A=\frac{c_A}{c}=\frac{n_A}{n}$

$x_B=$ mole fraction $of$ species $B=\frac{c_B}{c}=\frac{n_B}{n}$

$c=c_A+c_B$

$1=x_A+x_B$

Gas mixture

$c=\frac{n}{V}=\frac{P}{R}T$

$c_A=\frac{n_A}{V}=\frac{p_A}{R}T$

$c_B=\frac{n_B}{V}=\frac{p_B}{R}T$

$y_A=\frac{c_A}{c}=\frac{n_A}{n}=\frac{p_A}{p}$

$y_B=\frac{c_B}{c}=\frac{n_B}{n}=\frac{p_B}{p}$

$c=c_A+c_B=\frac{p_A}{RT}+\frac{p_B}{RT}=\frac{P}{RT}$

$1=y_A+y_B$

Please help ASAP