Answer: The required solutions are
[tex](x,y)=(3,2),~(-3,2),~(3,-2),~(-3, -2).[/tex]
Step-by-step explanation: We are given to solve the following system of equations:
[tex]4x^2+9y^2=72~~~~~~~~~~~~~~~~~~~~(i)\\\\x^2-y^2=5~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
Let us consider that
[tex]x^2=a,~~~~~y^2=b.[/tex]
So, equations (i) and (ii) becomes
[tex]4a+9b=72~~~~~~~~~~~~~~~~(iii)\\\\a-b=5~~~~~~~~~~~~~~~~~~~~(iv)[/tex]
Multiplying equation (iv) by 4 and then subtracting from equation (iii), we get
[tex](4a+9b)-(4a-4b)=72-4\times 5\\\\\Rightarrow 4a+9b-4a+4b=72-20\\\\\Rightarrow 13b=52\\\\\Rightarrow b=4.[/tex]
From equation (iv), we get
[tex]a-b=5\\\\\Rightarrow a-4=5\\\\\Rightarrow a=9.[/tex]
Therefore,
[tex]a=9~~~~~\Rightarrow x^2=9~~~~~\Rightarrow x=\pm3,\\\\b=4~~~~~\Rightarrow y^2=4~~~~~\Rightarrow y=\pm2.[/tex]
Thus, the required solutions are
[tex](x,y)=(3,2),~(-3,2),~(3, -2),~(-3, -2).[/tex]
