1.solve the equations below
x+y+z=1
x^2+y^2+z^2=35
x^3+y^3+z^3=97
2.Determine the area of the largest rectangle that can be inscribed in the circle x^2+y^2=a^2.
Thus, name the rectangle formed
3.a ship leaves airport and travels 25kmnon a bearing of 037° and then 60km on a bearing of 307°.calculate the distance from the port
4.The last digit of 2016^2017 is••••••••
5. Simplify 3√4^13?
One of the ways is to square and cube the first equation then substituting, but it gets pretty scary.
Lets use the "Just Look At It Theorem".
Look at x^2 + y^2 + z^2 = 35, assuming that there is an integer solution as there usually is in this type of question, all of the variables must be between -5 and +5
suppose x = 5, then 25 + y^2 + z^2 = 35, or y^2 + z^2 = 10
It should be easy to see that the only two squares that add up to 10 are 3^2 ans 1^2, thus y could be 3 and z = 1
So (5,3,1) would be a solution to the 2nd equation, (but so would (3,5,1) etc )
Can we make these work in the 1st ?
Mmmmhhhh, after about 20 seconds of consideration, it must be (5, -3, -1)
Well, look at this, that triple also works in the 3rd eqution:
5^3 + (-3)^3 + (-1)^3
= 125 - 27 - 1 = 97
Because of the symmetry of the equations, the 5,-3,-1 can be arranged in 3! ways, to have 6 possible solutions.
e.g. (-1,5,-3) would also work