Solving Simultaneous Equations for Confused Students

  • Context: High School 
  • Thread starter Thread starter Hypercase
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Discussion Overview

The discussion revolves around solving a pair of simultaneous equations involving variables raised to different powers. Participants explore methods for finding solutions, the nature of the equations, and the challenges associated with them. The scope includes mathematical reasoning and conceptual clarification regarding the complexity of the equations.

Discussion Character

  • Mathematical reasoning
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks help in solving the equations a1X^m + b1Y^n = c1 and a2X^m' + b2Y^n' = c2, expressing confusion about the notation.
  • Another participant questions the meaning of m' and suggests it might refer to the derivative of m.
  • A clarification is provided that m' indicates a different exponent for X in the second equation, leading to a rewritten form of the equations.
  • One participant proposes a method to isolate X in terms of Y, suggesting that the resulting equation could be solved numerically, although they express uncertainty about the next steps.
  • A later reply acknowledges that the equations do not simplify to a single variable and raises concerns about the complexity if the powers are large.
  • Another participant asks if there is a specific problem being addressed, while the original poster indicates they are considering the problem in a general sense.
  • Further discussion reveals that the variables X and Y are the focus, but a participant states that general solutions are not feasible due to the high degree of the resulting polynomial, suggesting numerical methods as an alternative.
  • One participant emphasizes the limitations of finding exact algebraic solutions for polynomials of degree greater than four, noting that real-life mathematics often involves ad hoc methods.

Areas of Agreement / Disagreement

Participants express varying opinions on the solvability of the equations, with some suggesting numerical methods while others highlight the limitations of finding general solutions. There is no consensus on a specific method or approach to solve the equations.

Contextual Notes

The discussion highlights the complexity of solving simultaneous equations with different powers and the limitations of algebraic methods for higher-degree polynomials. Participants acknowledge the need for numerical solutions but do not resolve the challenges presented.

Hypercase
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how can the following pair of simultaneous eqns. be solved

a1X^m +b1Y^n =c1
a2X^m’+b2Y^n’=c2
Please help:frown: [b(]
 
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What do you mean by m' ? Is it supposed to be the derivative of m?
 
No by m' I mean that the exponent of X in the first equation is not the same as the second equation.
Here, I've rewritten the equation:
a1X^m +b1Y^n =c1
a2X^p +b2Y^q =c2
 
So you have
[tex]a_1x^{n_1}+b_1y^{m_1}=c_1[/tex]
and
[tex]a_2x^{n_2}+b_2y^{m_2}=c_2[/tex]
Then
[tex]x=(\frac{c_2-b_2y^{m_2}}{a_2})^{\frac{1}{n_2}}[/tex]
Then you can substitute to get:
[tex]a_1(\frac{c_2-b_2y^{m_2}}{a_2})^{\frac{n_1}{n_2}} + b_1y^{m_1}=c_1[/tex]
which is an equation in a single variable. I'm not sure off the top of my head where to go from here, but that can definitely be solved numerically.
 
I reached had reached so far, but I noticed that this is not an equation in one variable(duh!). But if the powers were really big it would become a hell of a job. Is there another way or a easier way to proceed from here.
 
Is there a particular problem that you're trying to solve?
 
NO I'm thinking about this on a general basis.
 
Ok, so what are the variables, and what is known?
 
a1X^m +b1Y^n =c1
a2X^p +b2Y^q =c2

That is the eqn.
X,Y are the variables.
Now please help me solve it
 
  • #10
Sorry, but we can't help you. Not in general. You can rearrange all you want but, in general, you're going to end up with some high degree polynomial in one variable for which there are no general methods of solution (you can do up to degree 4 easily, degree 5 with a bit of ingenuity, but beyond that you're hoping for luck). I mean of course some nice exact algebraic solution. You could do it numerically and get approximate answers.

You can get various constraints on the solutions (are they integers etc) but the methods would be ad hoc.

Sorry, but that's what maths looks like in real life.
 

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