# Systems of algebraic equations

1. Dec 8, 2008

### jimmy1

If I had a linear system of algebraic equations, then I can relate the number of unknowns to the number of equations to determine if a solution exists. However, does this criteria carry over to nonlinear equations?

For example, I have a set of m>2 non linear equations and I have 2 unknowns. In general, is it possible to get a solution to this or even just show a solution exists? (I know if they were linear then no solution exists, but not sure with nonlinear eqns)

2. Dec 8, 2008

### HallsofIvy

Staff Emeritus
No it doesn't as, for example, the single equation x2= 4 in one "unknown" has two solutions while the single equation x2= 0 has exactly one solution and the single equation x2= -4 has no solutions. If it doesn't work for one equation it surely won't work for a system of equations!

3. Dec 8, 2008

### jimmy1

Ok, so is there any general method to show that a solution exists (or dosn't exist) to a system of non linear equations? What would the best approach be?

4. Dec 8, 2008

### mathman

If you have more equations than unknowns, there will in gnereal be no solutions. However, in some situations, the equations are not independent so you may have solutions. For example with 2 unknowns and more than 2 equations, try finding solutions using 2 equations and check to see if they satisfy the other equations.

jimmy1's comment is to a different question. When equations are non-linear, there may be mutiple solutions.

5. Dec 8, 2008

### jimmy1

So without finding any of these solutions, is there any way to show that they actually exist?

6. Dec 9, 2008

### mathman

In some cases, yes. However, there is no general approach that will work in all situations.

Correction in my previous note - I meant Halls of Ivy comment, not jimmy1.

7. Dec 10, 2008

### HallsofIvy

Staff Emeritus
mathman, in the original post, jimmy1 specifically asked about "non-linear" equations. I don't know why you said my comment "is to a different question".

8. Dec 10, 2008

### Office_Shredder

Staff Emeritus
The Inverse Function Theorem gives conditions on when a set of equations has a solution (assuming the equations are sufficiently differentiable) but doesn't tell you anything else

9. Dec 11, 2008

### mathman

My interpretation of his original question has is whether or not a system of several equations (m) in various unknowns (n), where n < m can be solved. I don't think he had any concern about multiple solutions to a single equation in one unknown.

10. Dec 12, 2008

### HallsofIvy

Staff Emeritus
And that was what I responded to- see my last sentence, " If it doesn't work for one equation it surely won't work for a system of equations!"