# Effect on beta one

#### statisticsisawesome

In linear regression, If the value of the x-variable changes, how do I provide a proof that the value of beta one will be affected?

#### Anonymous1

In linear regression, If the value of the x-variable changes, how do I provide a proof that the value of beta one will be affected?
Well the estimator is dependent on $$\displaystyle x_i:$$

$$\displaystyle \hat\beta_1 = \frac{\sum(x_i - \bar x)y_i}{\sum(x_i - \bar x)^2}$$

So...

#### statisticsisawesome

Is there a way to prove that mathematically?

#### Anonymous1

Is there a way to prove that mathematically?
Unless I am mistaken, this is trivially true since:

$$\displaystyle y = \beta_0 + \beta_1 x \implies \beta_1 = \frac{y-\beta_0}{x}$$

Can you think of a situation in which $$\displaystyle x$$ would change and $$\displaystyle \beta_1$$ wouldn't?