# Simultaneous non-linear equations

• HallsofIvy
In summary, your teacher wants you to find all positive integer solutions to the equation x+y=5. You can try to find the solutions by using a function or by direct computation. If x and y are both positive integers, then the only possibilities are x=1, x=2, x=3, x=4, and x=5.
HallsofIvy
Science Advisor
Homework Helper
abi ubong said:
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!

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 $$x^{x}+y^{y}=31$$

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

The obvious thing to do is write the first equation as y= 5- x and substitute into the second equation: xx+ (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!

Last edited by a moderator:
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:

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

#### Attachments

• roots.JPG
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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).

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

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

Last edited:
abia ubong: The "Lambert W function" is the inverse of the function f(x)= xex. 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 xx+ yy= 31?

Last edited by a moderator:
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

## 1. What are simultaneous non-linear equations?

Simultaneous non-linear equations are a set of equations that contain multiple variables and involve non-linear functions such as exponentials, logarithms, or trigonometric functions.

## 2. How do you solve simultaneous non-linear equations?

There is no single method for solving simultaneous non-linear equations, but some common techniques include substitution, elimination, and using a graphing calculator or computer program.

## 3. What makes simultaneous non-linear equations difficult to solve?

Simultaneous non-linear equations can be difficult to solve because there is no one set method for solving them, and they often involve complex mathematical operations. Additionally, they may have multiple solutions or no solutions at all.

## 4. How are simultaneous non-linear equations used in real life?

Simultaneous non-linear equations are used in a variety of fields, including physics, engineering, and economics, to model complex systems and to make predictions and solve problems.

## 5. Can simultaneous non-linear equations have more than two variables?

Yes, simultaneous non-linear equations can have any number of variables, although solving them becomes more difficult as the number of variables increases.

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