Take the following sample linear regression equation:

$$ \hat{y} = 20 + 50X $$

I understand how a change of units in X that consists of simple scalar multiplication would alter the regression equation. For example, if we suppose X is measured in millimeters and we change it to centimeters, then the equation would become:

$$\hat{y} = 20 + 500X$$

I do not understand how a linear transformation of $X$ that involves adding a constant would change the regression equation. For example, let us suppose that $X$ is measured in degrees Fahrenheit. How would changing to degrees Celsius change the equation? $C = \frac{5}{9}(F-32)$

Experimenting with R suggests that it would change both $\hat{\beta_0}$ and $\hat{\beta_1}$ but I do not understand how.

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    $\begingroup$ let $\hat{y}=b_0 + b_1x_1$ and substitute $x_2=cx_1+d$ into the first equation and simplify. $\endgroup$ – Glen_b Sep 24 '17 at 2:46

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