Question 45922
Hi Louis,

The solution isgiven in the problem itself. Always try to convert the statements of the problem into equations. Note -

Lets say the two numbers are x & y. Now as per the first statement, the sum of the squares of these numbers is 117. In mathematical language this would be written as:

{{{x^2 + y^2 = 117}}}

Lets put this as eq. no. 1

The second condition is that the difference of the squares is 45, therefore it can be written as:

{{{x^2 - y^2 = 45}}}

This gives us equation no. 2 ok ?
Now to solve a problem having two unknown quantities, we need atleast 2 sets of relations between them. Here we have the 2 relations as eq 1 & eq 2. Thus we may now proceed to solve them. In order to solve these equations, the approach is to manipulate them in such a manner that we are left with only one variable. Here if you simply add these two equations to each other, what we get is :

{{{x^2 + y^2 + x^2 - y^2 = 117 + 45}}}

I hope this much is clear

Or

{{{2x^2 = 162}}}

or

{{{x^2 = 81}}}    (divide both sides by 2)

Obviously x = 9.

Now put this value of x in eq 1, we get

{{{9^2 + y^2 = 117}}}

or

{{{81 + y^2 = 117}}}

or 

{{{y^2 = 117 - 81 = 36}}}

Thus y = 6.

Hence we have the solution as x= 9, y = 6. Please note that i have given detailed explanations which are not required when you actually solve the problem in an exam. Enjoy solving