+2729 Find all ordered pairs x, y of real numbers such that x + y = 10 and x^2 + y^2 = x^3 + y^3.
For example, to enter the solutions (2, 4) and (-3, 9), you would enter "(2,4),(-3,9)" (without the quotation marks).
+9675 Suppose xy=a. Then,
x2+y2=(x2+2xy+y2)−2xy=(x+y)2−2xy=100−2a
x3+y3=(x+y)3−3xy(x+y)=1000−30a
So the second equation is equivalent to
100−2a=1000−30a28a=900a=2257xy=2257x=2257y
Substituting this into x + y = 10, we have
y+2257y=107y2−70y+225=0
Checking the discriminant of this equation, Δ=(−70)2−4(7)(225)=−1400<0.
Therefore, there are no real solutions.
