Simple vs Multiple Linear Regression

In a simple linear regression model - we are only dealing with one IDV, such that:

y = b_{0} + b x

However - in most applications of OLS, we deal with more than 1 dependent variable for multiple linear regression, such that:

y = b_{0} + b_{1} x_{1} . . . + b_{n} x_{n}

When interpreting the effects of DV in response to multiple independent variables:

b0 - the y-intercept is the mean response of the DV when ALL IDV/predictor variables equal to 0 or at reference level for categorical variables.
$b_{n}$ - becomes the partial slope of the DV (y) and its corresponding $x_{n}$ - such that for every unit change in $x_{n}$ , Y changes by $b_{n}$ units - WHILE controlling for all other independent variables

Partial Slopes

Partial slopes refers to the amount of change in Y for a unit change in $x_{n}$ while controlling for all other independent variables

Note: that when x = 0, the predicted value becomes the y-intercept:

y = a + b_{1} x_{1} \to y = a + 0 \to y = a