please solve this equation step by step x^2-y^2=5 x^3-y^3=19

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nathanshields | High School Teacher | (Level 1) Associate Educator

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I'm not sure about this, but I'll think out loud and see if we can get it.

First thing to notice is that `x^2-y^2=(x+y)(x-y)` .  This is called "difference of squares" and is something you should memorize!  We can factor a difference of cubes too: `x^3-y^3 = (x-y)(x^2+xy+y^2)`

Now let's solve each equation for (x-y) and see what happens:

`x-y=5/(x+y)`

`x-y=19/(x^2+xy+y^2)`

Doesn't look promising, but let's set them equal and cross multiply:

`5/(x+y) = 19/(x^2+xy+y^2)`


`5x^2+5xy+5y^2=19x+19y`

 

Now, this is the equation of an ellipse, so there are infinitely many solutions (like (-19/15, 38/15)).  But the integer solutions are (0,0) (2,3) (3,2)

Check out http://www.wolframalpha.com/input/?i=5x^2%2B5xy%2B5y^2%3D19x%2B19y

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