5 - Algebra

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.

$x^2-xy=23$ 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 $x+y=45$ !