# What is the difference between linear equation and linear regression? [closed]

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?

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 $$\theta.$$

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.)

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.