We have to solve the simultaneous equations
x^2 + y^2=10 ...(1)
x^4 + y^4=82 ...(2)
let a = x^2 and b = y^2
This gives
a + b = 10 and a^2 + b^2 = 82
a + b = 10
=> a^2 + b^2 + 2ab = 100
substitute...
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We have to solve the simultaneous equations
x^2 + y^2=10 ...(1)
x^4 + y^4=82 ...(2)
let a = x^2 and b = y^2
This gives
a + b = 10 and a^2 + b^2 = 82
a + b = 10
=> a^2 + b^2 + 2ab = 100
substitute a^2 + b^2 = 82
=> 82 + 2ab = 100
=> 2ab = 18
=> ab = 9
substitute a = 10 - b
b(10 - b) = 9
=> b^2 - 10b = -9
=> b^2 - 9b - b + 9 = 0
=> b(b - 9) - 1(b - 9) = 0
=> (b - 1)(b - 9) = 0
=> b = 1 and b = 9
For b = 1, a = 9
for b = 9, a = 1
Now b = y^2 = 1 => b = -1, + 1 and a = x^2 = 3 , -3
y^2 = 9 => y = -3 and +3 and x^2 = 1 => x = -1 and +1
The required solutions are: (3, -1), (3 , 1), (-3 , -1), (-3, 1), (1 , -3), (1 , 3) ,(-1 , 3) and (-1 , -3)