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I am using linear equation to find the correlation of values of x, y and z

3x + 4y + 2z = 6
4x + 1y + 4z = 17
8x + 2y + 5z = 9

I would like to know the difference between linear equation and linear regression. Is it the same in approach?

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It's the viewpoint that makes the difference. A linear equation is one in which the variables show up in a linear fashion. So your $x$'s, $y$'s, and $z$'s, etc., aren't raised to powers, don't show up in functions like $\sin(x),$ etc.

A linear regression is one in which the coefficients show up in a linear fashion. So if you were trying to fit the equation $y=ae^{x}$ to some data, your goal is to find $a,$ which shows up linearly. It is therefore a linear regression, even if it's not a linear equation in $x.$

However, if you were to try to regress $y=A\cos(\omega t+\theta),$ where you're trying to find $A,\omega,$ and $\theta,$ it would not be a linear regression problem because the coefficients you're trying to find don't show up linearly. That is, finding $A$ is linear, but not $\omega$ or $\theta.$

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The essential difference is that an algebraic equation has no random (stochastic) component.

However, for a regression model, for example:

$$Y = \beta_0 + \beta_1 X_0 + \beta_2 X_1 + \beta_2 X_2 + \epsilon$$

Where the explanatory variables are presumed to be fixed and known (non-stochastic in nature) together with a single random error term.

In the special case of a Least-Absolute Deviation (LAD) regression model, the error term is presumed to follow a Laplace distribution, while for classic Least-Squares regression, the error term is presumed to be a Normal random deviate with a mean value of zero.

Wikipedia on the Logistic regression further comments:

The logistic regression can be understood simply as finding the $\beta $ parameters that best fit:

$y = 1 \text{ } \beta_0 + \beta_1 x + \epsilon > 0 $

$\text{ } = 0$ elsewhere

$\epsilon $ is an error distributed by the standard logistic distribution. (If the standard normal distribution is used instead, it is a probit model.)

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In your example, you have three equations and three unknowns so you can solve for x, y and z and correlation will make no sense. Also, this would then be a math question and not a statistics one.

Correlation of variables only makes sense when there are multiple observations (you don't seem to have observations at all) and it usually involves some stochastic element.

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