Mathematics Question on solution of system of linear inequalities in two variables

Question

The number of ordered pairs (x, y) satisfying ${3^{x} . 5^{y} = 75}$ and ${3^{y} . 5^{x} = 45}$ is

Answer 1

Solution

The equation are ${3^{x} . 5^{y} = 75}$ ...(1) and ${3^{y} . 5^{x} = 45}$ ....(2) Dividing the two equations, we get $\left( \frac{3}{5} \right) {^{x -y} = \frac{75}{45} = \frac{5}{3}$ $\Rightarrow x - y = - 1}$ .....(3) Multiplying equations (1) and (2), we get ${ (15)^{x + y} = 45 \times 75 = (15)^3$ $\Rightarrow x + y = 3}$ ....(4) solving (3) and (4) we get ${x = 1, y = 2}$