How to understand this multiple regression question without having an example in the textbook?

Question

How to understand this multiple regression question without having an example in the textbook?

Please briefly show how to do the final question and give the answer to the final question.

Answer 1

A point about the formula for $Y_i$: $u_i$ is the residual for case $i$ and is the difference between the observed and estimated values. (Diagnostics can be performed on the residuals to verify that a linear multiple regression is appropriate).

For the estimated value,

$$\tilde{y}=\beta_0+\beta_1 x_1 + \beta_2 x_2$$

and this equation is linear, so for changes in values $\Delta \tilde{y}=\beta_1 \Delta x_1 + \beta_2 \Delta x_2$.

Then for (a)

$$\Delta\tilde{y}=3\beta_1$$ and for (b) $$\Delta\tilde{y}=-5\beta_2$$ so by linearity, for (c): $$\Delta\tilde{y}=3\beta_1-5\beta_2$$

Answer 2

Y is basically just a function at this point, and X1 and X2 are its input parameters. You can then treat the changes in X1 and X2 as numerical differences.
For the first part, you could say Y = B0 + B1*(X1+3) + B2*X2 (ignoring the error term, because we can never know what the error term will be). Compared to the original value, this new Y is 3*B1 larger. Similarly, for part two, it will be 5*B2 smaller.
For the last part, we combine the two answers and say that it's (3*B1)-(5*B2) larger.