SOLUTION: Hi again, This type of problem is still giving me some trouble: The sum of the squares of two numbers is 117. The difference of the squares of the same two numbers is 45. Fin

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Question 45922: Hi again,
This type of problem is still giving me some trouble:
The sum of the squares of two numbers is 117. The difference of the squares of the same two numbers is 45. Find the numbers.
Thanks again,
Louis
Answer by abhijitvakil(7) (Show Source):
You can put this solution on YOUR website!
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:

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:

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 :

I hope this much is clear
Or

or
(divide both sides by 2)
Obviously x = 9.
Now put this value of x in eq 1, we get

or

or

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