Systems of algebraic equations

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In summary, the conversation discusses the criteria for determining the existence of solutions in a linear system of algebraic equations and whether this carries over to nonlinear equations. It is mentioned that if there are more equations than unknowns, there will generally be no solutions, but in some cases, there may be multiple solutions. The possibility of showing the existence of solutions without finding them is also mentioned, but there is no general approach for this. The Inverse Function Theorem is brought up as a potential tool for finding solutions, but it does not provide any other information.
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jimmy1
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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)
 
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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
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
jimmy1 said:
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)


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
mathman said:
When equations are non-linear, there may be mutiple solutions.

So without finding any of these solutions, is there any way to show that they actually exist?
 
  • #6
jimmy1 said:
So without finding any of these solutions, is there any way to show that they actually exist?

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
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
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
HallsofIvy said:
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".

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
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!"
 

FAQ: Systems of algebraic equations

What is a system of algebraic equations?

A system of algebraic equations is a set of two or more equations that contain two or more variables. The goal is to find values for the variables that satisfy all of the equations simultaneously.

How do you solve a system of algebraic equations?

There are several methods for solving a system of algebraic equations, including substitution, elimination, and graphing. The most common method is substitution, where one equation is solved for one variable and then that value is substituted into the other equations.

Can a system of algebraic equations have more than one solution?

Yes, a system of algebraic equations can have one, none, or infinitely many solutions. This depends on the number of variables and equations in the system and how they relate to each other.

What is the difference between consistent and inconsistent systems of algebraic equations?

A consistent system of algebraic equations has at least one solution that satisfies all of the equations, while an inconsistent system has no solutions that satisfy all of the equations. This can be determined by graphing or using elimination to see if the equations intersect at one point or are parallel lines.

How are systems of algebraic equations used in real life?

Systems of algebraic equations are used in a variety of real-life situations, such as in business, science, and engineering. They can be used to model and solve problems involving multiple variables, such as finding the optimal solution to a manufacturing problem or predicting the trajectory of a projectile.

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