solve non-linear system: x^(x + y) = y^(x - y), x^2 * y = 1
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mmm4444bot
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The second equation tells us that y is the reciprocal of x^2.
Since 1 is it's own reciprocal, we find our first solution by inspection.
x = 1
y = 1
Substituting 1/x^2 for y in the first equation leads to the following.
x^(x^3 + 1) = (x^2)^(1 - x^3)
Equating exponents leads to the following.
3x^3 - 1 = 0
This has one Real solution, and it provides a second solution to the original system.
x = cuberoot(9)/3
y = cuberoot(9)
Since 1 is it's own reciprocal, we find our first solution by inspection.
x = 1
y = 1
Substituting 1/x^2 for y in the first equation leads to the following.
x^(x^3 + 1) = (x^2)^(1 - x^3)
Equating exponents leads to the following.
3x^3 - 1 = 0
This has one Real solution, and it provides a second solution to the original system.
x = cuberoot(9)/3
y = cuberoot(9)