Solution for a polynomial system of three equations

In summary, The system of equations given is solved by using the basic principles of systems of equations, eliminating one unknown at a time until you have a single equation in one unknown. This method can be applied to any kind of system, including the one provided. The solution involves solving a quadratic equation, which can be found by using the quadratic formula, and then back substituting to find the solutions for a, b, and c in terms of B', C', and D'.
  • #1
Bruno Tolentino
97
0
I desire to know tho solution (solution for a, b and c in terms of B', C' and D') of this system of equation:

B' = - 2 a - c
C' = 2 a c + b²
D' = - b² c

I don't know none method for solve this kind of system, therefore, I came to ask here...
 
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  • #2
Hi Bruno,

What is the context of this question? Is this for schoolwork?
 
  • #3
Bruno,
Since time is short, I ran the problem without any prior tricks through Mathematica. If you know MMA, I naively entered the equations you provided into it's Solve[ ] function. The symbolic answer is a very, very ugly set. This kind of brute force, followed by some simplification, gives 6 solution sets for the 3 variables. When I substituted B=1, C=1, D=1 into these, there were 2 sets of real solutions. The other 4 were complex solutions. In some cases, one of the variables = 0.

If you are trying to find the answer to some coursework, then I suggest you find some trick that radically simplifies the equations. I'd start by solving one of these for one variable in terms of all others and substitute into the remaining eqns. Play around a bit. There's probably some symmetry that makes the problem radically simple. Good luck.
 
  • #4
Why are you under the impression that you have to learn a different "method" for any new kind of system? The basic ideas of "systems of equations" work nicely here. Your equations are B' = - 2 a - c, C' = 2 a c + b², and D' = - b² c. "Eliminate" one unknown at a time, just as you would for systems of linear equations until you have a single equation in one unknown. From the last equation, b²= -D'/c. Substituting into the second equation, C'= -2ac+ D'c. So 2ac= D'c- C' and a= (D'c- C')/2c. Then the first equation becomes B'= (C'- D'c)/c- c so B'+ c= (C'- D'c)/c, (B'+ c)c= c²+ B'c= C'- D'c, c²+ (B'+ D')c- C'= 0. That is a quadratic equation in c. Use the quadratic formula to find two solutions to that, the "back substitute" into a= (D'c- C')2c and b²= -D'/c to find corresponding solutions for and b.
 
  • #5
Bruno Tolentino said:
I desire to know tho solution (solution for a, b and c in terms of B', C' and D') of this system of equation:

B' = - 2 a - c
C' = 2 a c + b²
D' = - b² c

I don't know none method for solve this kind of system, therefore, I came to ask here...

I would advise you to study elimination theory and Gröbner bases.
 

1. What is a polynomial system of three equations?

A polynomial system of three equations is a set of three equations that involve polynomial expressions. These equations can be linear, quadratic, cubic, or any other degree, and they are usually solved simultaneously to find the values of the unknown variables that satisfy all three equations.

2. How do you solve a polynomial system of three equations?

To solve a polynomial system of three equations, you can use various methods such as substitution, elimination, or graphing. Substitution involves solving one equation for one variable and then plugging that value into the other equations. Elimination involves combining the equations to eliminate one variable and then solving for the remaining variables. Graphing involves plotting the equations on a graph and finding the intersection points.

3. Can a polynomial system of three equations have more than one solution?

Yes, a polynomial system of three equations can have more than one solution. This means that there can be multiple sets of values for the unknown variables that satisfy all three equations. These solutions can be real or complex numbers, depending on the degree and coefficients of the equations.

4. What is the importance of solving a polynomial system of three equations?

The solution of a polynomial system of three equations can be used to find the values of unknown variables in various real-life situations. For example, it can be used to solve for the coordinates of intersection points in geometry problems or to find the roots of a polynomial function in algebraic equations.

5. Can a polynomial system of three equations be solved for any degree of polynomials?

Yes, a polynomial system of three equations can be solved for any degree of polynomials. However, as the degree increases, the complexity of the equations and the difficulty of finding solutions also increases. Advanced mathematical techniques such as matrix operations and polynomial division may be required to solve higher-degree polynomial systems.

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