# Determining coefficients from an equation with 3 variables

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• RichardWattUK
In summary, the author is trying to solve for the contact characteristics of a recess action worm gear drive with double-depth teeth, and he's looking for a way to calculate the coefficients for Cramer's Rule. He has some questions about the notation and the algebraic structure of the equation.
RichardWattUK
TL;DR Summary
Determining coefficients from an equation with 3 variables
Hi,

Some of the background related to this question is in this thread, but I've got another question as I'm looking at another problem that has come up with the same application which I'm trying to solve using the equation of meshing for a worm gear and the cutting/grinding tool that creates it. This is taken from the paper "Contact Characteristics of Recess Action Worm Gear Drives With Double-Depth Teeth", which you can view here, specifically Equation 16:

$$f(l_1,\theta_1,\phi_1(\phi_2))=\omega_1{[(m_21 cos \gamma_1-1) Y_1+m_21(cos \phi_1 sin \gamma_1 Z_1 + sin \phi_1 cos \gamma_1 C_1)]N_x1+[-(m_21 sin \gamma_1-1) X_1+m_21(-sin \phi_1 sin \gamma_1 Z_1 + cos \phi_1 cos \gamma_1 C_1)]N_y1+[m_21 sin \gamma_1(-cos \phi_1 X_1 + sin \phi_1 Y_1 + C_1)]N_z1}=0$$

This has 3 variables in it - ##l_1##, ##\theta_1##, ##\phi_1(\phi_2)## - now, some of the notation is not familiar to me since it's been about 20 years since I last studied math(s), but I've found that I may be able to use Cramer's Rule to solve for the 3 variables if I can construct a 3x3 matrix of the coefficients and a 3x1 vector for the results.

What I also find strange is there's the ##f(l_1,\theta_1,\phi_1(\phi_2))## part but the main equation only references ##\theta_1##, but since this equation is created using other equations, that could cause the loss of ##l_1## and ##\theta_1## due to the substitutions and expansions I suppose?

So, how would I solve this equation, and how would I get the coefficients from it to use with Cramer's Rule? It looks to me like a matrix system of linear equations, but is it really?

The expression doesn't look right. ##\phi_1## is the only variable there.

mathman said:
The expression doesn't look right. ##\phi_1## is the only variable there.
I was thinking the same thing, but that's how it's written in the paper and I've been looking for an errata in case there were any corrections made since but I can't find anything.

It looks there may be typos like ##\gamma_1=\theta_1?## and ##1=l_1?##.

## What is the purpose of determining coefficients from an equation with 3 variables?

The purpose of determining coefficients from an equation with 3 variables is to understand the relationship between the variables and how they affect each other. It allows us to make predictions and solve complex problems in various fields such as physics, chemistry, and economics.

## How do you determine the coefficients from an equation with 3 variables?

To determine the coefficients from an equation with 3 variables, you can use techniques such as substitution, elimination, or matrix operations. These methods involve manipulating the equation to isolate one variable at a time and then solving for its coefficient.

## What are the common challenges when determining coefficients from an equation with 3 variables?

One of the common challenges is dealing with non-linear equations, where the variables are raised to different powers. Another challenge is when the coefficients are fractions or decimals, which can make the calculations more complex. Additionally, if the equation is not in standard form, it may require extra steps to determine the coefficients.

## Can you determine the coefficients from an equation with 3 variables using technology?

Yes, there are various software and online tools available that can help determine the coefficients from an equation with 3 variables. These tools use algorithms and numerical methods to solve the equation and provide the coefficients. However, it is still essential to understand the concepts and methods behind determining coefficients manually.

## What are some real-life applications of determining coefficients from an equation with 3 variables?

Determining coefficients from an equation with 3 variables has many real-life applications, such as predicting the growth of a population, analyzing chemical reactions, and optimizing production processes in industries. It is also used in fields like economics to study the relationship between different factors, such as supply and demand, and make informed decisions.

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