How do I solve x^3 + y^3 = 1, x^4 + y^4 = 1

[mohlam12]

hey everyone, here is my problem, u have to solve this:
x^3 + y^3 = 1
x^4 + y^4 = 1

i tried to solve it but i couldnt, please just give me some hints if you can, thx

i had this one too,
2x+xy+2y = 59
3x-2xy+3y = -34

the teacher told me to do:
2(x+y)+xy=59
3(x+y)-2xy=-34

then,
x+y=12
xy=34

then i solved the equation : x^2-12x+35=0
i got x=5 and x=7
so, the solutions are
x=5 y=7 and x=7 and y=5

i didn t get it, he explained to me but i still don t get why, can anyone explain to me how it was solved?

 

[dextercioby]

For the first,here's what my maple had to say:
Solution is : $\left\{ y=0,x=1\right\} ,\allowbreak \left\{ y=1,x=0\right\} ,\allowbreak \left\{ y=-2\rho ^4-4\rho ^3-6\rho ^2-5\rho -3,x=\rho \right\} $ where $\rho $ is a root of $2Z^6+6Z^5+12Z^4+16Z^3+15Z^2+9Z+4$

,which,translated,means:
Solution is \{y=0,x=1\},\{y=1,x=0\},\{y=-2\rho^{4}-4\rho^{3}-6\rho^{2}-5\rho-3,x=\rho\} \where \ [\tex] <br /> [\tex] \rho \ is \ a \ root \ of \ 2z^{6}+6z^{5}+12z^{4}+16z^{3}+9z+4

Daniel.

 

[dextercioby]

Yes,those substitutions turn the initial system into one which is simpler.There's not too much to explain,though,these things are working only with particular examples,you can generalize in any way.

It's more of an inspiration,really.You either have it (case in which u see the trick you have to pull),or not.

Daniel.

 

[mohlam12]

For the first,here's what my maple had to say:
Solution is : $\left\{ y=0,x=1\right\} ,\allowbreak \left\{ y=1,x=0\right\} ,\allowbreak \left\{ y=-2\rho ^4-4\rho ^3-6\rho ^2-5\rho -3,x=\rho \right\} $ where $\rho $ is a root of $2Z^6+6Z^5+12Z^4+16Z^3+15Z^2+9Z+4$

,which,translated,means:
Solution is \{y=0,x=1\},\{y=1,x=0\},\{y=-2\rho^{4}-4\rho^{3}-6\rho^{2}-5\rho-3,x=\rho\} \where \ [\tex] <br /> [\tex] \rho \ is \ a \ root \ of \ 2z^{6}+6z^{5}+12z^{4}+16z^{3}+9z+4

Daniel.

and...do u know how to get {0,1} {1,0} ... without ur maple ?

 

[dextercioby]

You can see it directly that,y chosing x=0,then "y" is a solution of the system:
y^{3}=1;y^{4}=1,which is of course y=1.
The same if you chose y=0,you'll find x=1,that's because the system is homogenous in "x" and "y".

Daniel.