Simple question about linear regression

Question

I am dealing with the following exercise:

Exercise. In a linear regression model of the form $y = w_1x + w_0$, if we increase the value of $x$ by one unit, what can we expect from the value of of $y$?

$(a)$ An increase of value equal to $1$;

$(b)$ An increase of value equal to $w_0$;

$(c)$ An increase of value equal to $w_1;$

$(d)$ It is impossible to tell anything about the increase on $y$.

My attempt. As an aspirant to be a mathematician, this question leads to one simple line of thinking: if $x$ is increased by one unit, one might think of this as a simple variable change of the form $x \to x+1.$ If this is case, we obviously expect $y$ to suffer an increase of value equal to the parameter $w_1,$ hence I would select option $(c).$

My concerns. After some further thought, what exactly is "one unit" of the variable $x$? As far as I am concerned, this might not be a concept as objeticve as I thought. This makes me think that probably option $(d)$ makes some sense aswell.

All the information I have about the question is posted in the Exercise. section. Thanks for any help in advance.

Answer 1

The answer to the question is certainly $(c)$. With a one unit increase in $x$, you simply multiply this value by the slope value, and this will yield an increase in $y$. So if the equation is:

$$ y = .54x + 45 $$

This would simplify to an increase of value equal to the slope (the $w_1$ in your question):

$$ y = .54 + 45 $$

Resulting in this $y$, a simple $.54$ addition to the intercept:

$$ y = 45.54 $$

To your question about what a one-unit increase in $x$ actually means, I suppose that depends on the measurement of the $x$ (centimeters, pounds, etc.). However, that doesn't seem to matter here. The units counted are not the same as the type of unit involved. Therefore the answer would still be $(c)$.