3 Equations with 2 Unknowns

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In summary, the conversation discusses 3 equations with two unknowns, r_i == f_i (g, \beta), where g and \beta are independent variables and r_i are known experimentally with associated errors. The equations are variations of r_1 = \frac{(g^2 x + g^2 y - \frac{\beta}{3})^2}{g^2}. The question is how to solve for the best fit of the two parameters, but a definition of "best" must be provided. Some suggestions include using statistics and data, or defining the best value of the unknowns as the one that gives the best least squares fit when plotted against the data used to estimate the constants. It is recommended to use
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Hepth

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I have 3 equations with two unknowns of the form:

[tex]r_i == f_i (g,\beta)[/tex]

Where g and [itex]\beta[/itex] are the independent unknown variables and [itex]r_i[/itex] are known experimentally, but have some error associated with them, say something like [itex]4.3 \pm 0.3[/itex] and there are other errors associated with constants inside the [itex]f_i[/itex].

My question is how do I solve for the objectively "best" fit to those two parameters. I remember doing something like this back in prob/stats, I believe I do something like a [itex]\chi^2[/itex] fit but I can't find anything online about it that doesn't concern bins of data points, but rather continuous functions. (Or do I have to simulate data, then fit it)

Can anyone lead me in the right direction? I don't have any of my undergrad books with me.

The equations are all something like :
[tex]
r_1 =\frac{\left(g^2 x+g^2y-\frac{\beta }{3}\right)^2}{g^2 }
[/tex]
with some variations on the form. So nothing with unsolvable functions.

EDIT: Oh and I plan on using Mathematica to do this, but I'd rather understand the mathematics first.
 
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Hepth said:
My question is how do I solve for the objectively "best" fit to those two parameters. I remember doing something like this back in prob/stats,

There is no mathematical definition for "objectively best fit", so you have to supply the details. (You aren't alone. If you browse the posts in this section of the forum you find person after person asking for the "best" solution to problems. Most are not interested enough to supply a definition of "best" and a few seem downright offended to be asked for it.)

If you remember something from statistics, it probably involved data. Do you have the data that was used to supply the "known" parameters in these equations? If you don't have the data, do you know something about its distribution?

I can imagine a solution based on an arbitrary definition for "best fit" and imagining that you have data. You can define the best value of the unknowns as the one gives the best least squares fit of the equations when they are plotted versus the data that was used to estimate the constants in the equations. However, you'll get better advice if you use all the details of the real world problem that you are trying to solve, rather than solving some abstract carciature of it.
 

1. What are "3 Equations with 2 Unknowns"?

"3 Equations with 2 Unknowns" refers to a system of three linear equations with two variables that need to be solved simultaneously.

2. How do you solve a system of "3 Equations with 2 Unknowns"?

To solve a system of "3 Equations with 2 Unknowns", you can use methods such as substitution, elimination, or graphing. These methods involve manipulating the equations to eliminate one variable and solve for the other.

3. Why do we use "3 Equations with 2 Unknowns"?

"3 Equations with 2 Unknowns" are used to model real-world situations where there are three different relationships between two variables. By solving the system, we can find the values of the variables that satisfy all three equations and better understand the relationship between them.

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Solving "3 Equations with 2 Unknowns" is important in fields such as science, engineering, and economics where there are often multiple variables that are interrelated. By solving the system, we can make predictions and analyze the relationships between these variables.

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Yes, a system of "3 Equations with 2 Unknowns" can have one, zero, or infinitely many solutions. The number of solutions depends on the relationships between the equations and the variables involved.

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