If (x³ - y³) : (x² + xy + y²) = 5 : 1 and (x² - y²) : (x - y) = 7 : 1, then the ratio 2x : 3y equals = ?

Solution(By Examveda Team)

According to question,
$$\eqalign{ & \frac{{{x^3} - {y^3}}}{{{x^2} + xy + {y^2}}} = \frac{5}{1} \cr & \frac{{\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)}}{{{x^2} + xy + {y^2}}} = \frac{5}{1} \cr & x - y = 5.....({\text{i}}) \cr & \frac{{{x^2} - {y^2}}}{{x - y}} = \frac{7}{1} \cr & \frac{{\left( {x + y} \right)\left( {x - y} \right)}}{{x - y}} = \frac{7}{1} \cr & x + y = 7.....({\text{ii}}) \cr & {\text{Solve equations (i) and (ii)}} \cr & x = 6 \cr & y = 1 \cr & \therefore \frac{{2x}}{{3y}} = \frac{{2 \times 6}}{{3 \times 1}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{12}}{3} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{4}{1} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4:1 \cr} $$