5 - Algebra

The numbers x x and y y are positive integers where x 2 x y = 23 x^2-xy=23 . What is the value of x + y x+y ?

34 35 30 24 45

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2 solutions

Farah Roslend
Jun 18, 2015

Lol I sort of cheated by using the values from the options given, and a calculator.

What I did was:

Let x+y = k ....(1)

x^2 -xy=23

x- 23/x = y .....(2)

(1)+(2): 2x^2-kx-23=0 x=[k +/- sqrt(k^2+184)]/4

Since sqrt(k^2+184) > sqrt(k^2), thus consider only x=[k + sqrt(k^2+184)]/4, since k-sqrt(k^2+184)<0 because x +y=k>0 since both x and y are positive.

Now, using the values given in the options and a calculator, k=30, 24,35,34 gives an decimal number while k=45 gives an integer.

Since x and y are integers, k should be an integer, so the answer must be 45.

Parth Bhardwaj
Jun 17, 2015

x 2 x y = 23 x^2-xy=23 can be factorized as - x(x-y)=23.Which tells us that product of x and (x-y) would be 23. And as we know that 23 has no other factors other than 1 and itself ( its a prime number).Now we conclude that x=23 and (x-y)=1 (we cant say that x=1 and x-y=23 , because x-y is surely less than x, as both of them are positive integers) When x=23, we get the value of y as 22 (from the equation x-y=1). Thus x + y = 45 x+y=45 !

Moderator note:

Given the negative signs, you need to be careful with your explanation. It could also use some rephrasing for clarity, as it appears you have gotten mixed up.

In the end, you stated that x = 23 , y = 22 x = 23, y = 22 , but then x + y = 45 x + y = 45 wouldn't it?

I think there a typo...


Rishabh Tripathi - 5 years, 12 months ago

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