SOLUTION: if x+y=3 and x^2+y^2=6, find the numerical value of x^3+y^3

Algebra ->  Finance -> SOLUTION: if x+y=3 and x^2+y^2=6, find the numerical value of x^3+y^3      Log On


   

Question 1091393: if x+y=3 and x^2+y^2=6, find the numerical value of x^3+y^3
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

This is a type of algebra problem commonly encountered on high school math contests.

You need to use the given equations somehow to get an expression containing x^3 and y^3. One way to do this is to cube the first equation:

%28x%2By%29%5E3+=+x%5E3+%2B+3%28x%5E2%29y+%2B+3xy%5E2+%2B+y%5E3
%28x%2By%29%5E3+=+%28x%5E3+%2B+y%5E3%29+%2B+3xy%28x%2By%29
x%5E3+%2B+y%5E3+=+%28x%2By%29%5E3+-+3xy%28x%2By%29

You know the value of (x+y); to get a numerical value for this expression, you need to find the value of xy. To do that, you can square the first equation:

%28x%2By%29%5E2+=+x%5E2+%2B+2xy+%2B+y%5E2+=+%28x%5E2+%2B+y%5E2%29+%2B+2xy+=+9
6+%2B+2xy+=+9
2xy+=+3
xy+=+3%2F2

And finally


Another way to get an expression that includes x^3 and y^3 is to multiply the two given equations together.

x%5E3+%2B+y%5E3+=+%28x%2By%29%28x%5E2%2By%5E2%29+-+xy%28x%2By%29
x%5E3+%2B+y%5E3+=+%283%29%286%29+-+%283%2F2%29%283%29+=+18-9%2F2+=+27%2F2