Question

Solve the following system. 4x2 + 9y2 =72 x2 - y2 = 5 The solutions are (), (), (), and () (remember to include the commas) there has to be 4 solutions all ordered pairs please help

Answer 1

Answer: The required solutions are

`(x,y)=(3,2),~(-3,2),~(3,-2),~(-3, -2).`

Step-by-step explanation: We are given to solve the following system of equations:

`4x^2+9y^2=72~~~~~~~~~~~~~~~~~~~~(i)\\\\x^2-y^2=5~~~~~~~~~~~~~~~~~~~~~~~~(ii)`

Let us consider that

`x^2=a,~~~~~y^2=b.`

So, equations (i) and (ii) becomes

`4a+9b=72~~~~~~~~~~~~~~~~(iii)\\\\a-b=5~~~~~~~~~~~~~~~~~~~~(iv)`

Multiplying equation (iv) by 4 and then subtracting from equation (iii), we get

`(4a+9b)-(4a-4b)=72-4* 5\\\\\Rightarrow 4a+9b-4a+4b=72-20\\\\\Rightarrow 13b=52\\\\\Rightarrow b=4.`

From equation (iv), we get

`a-b=5\\\\\Rightarrow a-4=5\\\\\Rightarrow a=9.`

Therefore,

`a=9~~~~~\Rightarrow x^2=9~~~~~\Rightarrow x=\pm3,\\\\b=4~~~~~\Rightarrow y^2=4~~~~~\Rightarrow y=\pm2.`

Thus, the required solutions are

`(x,y)=(3,2),~(-3,2),~(3, -2),~(-3, -2).`