Simultaneous non-linear equations

[HallsofIvy]

hey i was given this at school anD can't do it its a simul eqn. x+y=5,x^x+y^y=31.

This was sent to me as a personal message- it's always better to post questions like this than just send them to me!

 

[arildno]

abi ubong, hint:
Try to find all positve integer solutions to your system;
that is what I think your teacher is after!

1.So, your first job is to determine:
Which pairs of positive integers (x,y) satisfies the equation: x+y=5

2. Then, you must determine: Which of those pairs you found under 1. also satisfy $$x^{x}+y^{y}=31$$

 

[HallsofIvy]

Are you certain you copied this correctly? x+y= 5, x²+ y²= 31 would be very easy, x+ y= 5, x^x+ y^y= 31 is very hard!

The obvious thing to do is write the first equation as y= 5- x and substitute into the second equation: x^x+ (5-x)^5-x= 31. You might be able to put this into a form you could apply the "Lambert W function" to, but that's certainly not elementary. Other than that, I would suggest a numerical solution.

Arildno got in before me- his suggestion: look for positive integer solutions reduces the possible solutions so it can be done by direct computation.
Assuming, of course, that there are positive integer solutions!

 

[saltydog]

I aint' proud. Am I helping out too much? Really, I didnt' even know to check for integers until I gave up and scrolled down to Arildno's post. Anyway, the plot is for:

$$y(x)=x^x+(5-x)^{5-x}-31$$

 

[arildno]

Just to add something in regard to saltydog's post (I hope he'll agree with me on this):
abi ubong:
The roots of saltydog's FUNCTION Y(x) will give you the NUMBERS x in the NUMBER PAIRS (x,y) which are solutions to your system.
The corresponding NUMBERS y is found by the equation y=5-x, where x is a root for Y(x).

 

[abia ubong]

i still do not get any of this especially urs hallsofivy wats rthe lambert w function or watever it is ,plssssssssssssss i need graet help

 

[arildno]

Well, since you evidently haven't bothered to read my reply to you, don't expect anymore help on this.

 

[HallsofIvy]

abia ubong: The "Lambert W function" is the inverse of the function f(x)= xe^x. It can be used to solve many equations in which x is both an exponent and a base.

However, if you are not in college, you probably would not be expected to know that function. Look at "salty dog"'s and "arildno"'s responses!

Look again at arildno's first suggestion. If x and y have to be positive integers AND their sum is 5, the only possibilities are:
x= 1, y= 4
x= 2, y= 3
x= 3, y= 2
x= 4, y= 1

Do any of those satisfy x^x+ y^y= 31?

 

[abia ubong]

hey firstly do not get offended arildno,also hallsofivy wat if x or y or maybe both were 2 be negative,such methods might not work i need a general solution .
thnxs