Find one pair of real numbers, (x,y), such that x + y = 6 and x^3 + y^3 = 144.

Find one pair of real numbers, (x,y), such that x + y = 6 and x^3 + y^3 = 144.

x=5.2361....  and y=.7639.....

x + y = 6 and x^3 + y^3 = 144

Notice that   y = 6 - x   and subtituting this into the second equation, we have

x^3  + [6 - x]^3 = 144  simpify

x^3  -x^3+18 x^2-108 x+216  = 144

18x^2 - 108x + 216 = 144  subtract 144 from each side

18x^2 - 108x +72  = 0    divide through by 18

x^2 - 6x + 4 = 0    and using the Quadratic Formula, the solutions for this are...x =3 +sqrt(5) and x = 3 - sqrt(5)

So..if we choose that x = 3 + sqrt(5), then y =  6  - [3 + sqrt(5)] = 3 - sqrt(5) 

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