What do we mean by linear?

The word linear relates to lines. In one variable, say x, a linear function has the familiar form y = mx + b. It's graph is a line with slope m and a y-intercept of b. A linear function of n variables, say $x_1, x_2, \dots, x_n$, a linear function has the form

$$z = f(x_1, x_2, \dots, x_n) = a_1 x_1 + a_2 x_2 + \dots + a_n x_n + b.$$

The constants $a_1, \dots, a_n$ are the slopes of the function in each direction. Thus, if we fix all the variables except x₁ and then change x₁ by one unit, the total change in z will be $\Delta z = a_1 \Delta x = a_1$. Notice that, as in the one variable case, each variable appears only to the first power in a polynomial. No other functional dependence on the variables is allowed if the function is to be linear. We cannot have cross terms either, like x₁ x₂.