## What is a linear function of several independent variables?

In section 1.1, we introduced a linear function of one variable, *y* = *mx* + *b*. There was nothing special about the names of the variables, *x* and *y*, or the names of the constants, *m* and *b*. Another possible form for a linear function of *x* and *y* is *y* = *a*_{0} + *a*_{1 }*x*. In this format, a_{0} is the vertical intercept and a_{1} is the slope.

When several independent variables are introduced, it is prudent to use names for the variables that make sense. If one variable is named *x*, we can extend this to *n* variables using subscripts. Subscripts are numbers that appear to the right of the variable and slightly lowered. The subscript is a part of the variable’s name and is useful to show generically that there are many variables. For instance, if we wanted to define a function with three independent variables that describe the quantities of three different products, we might use *Q*_{1}, *Q*_{2}, and *Q*_{3}.

In general, let *x*_{1}, *x*_{2}, … , *x*_{n} be the names of *n* independent variables.

A linear function of *n* independent variables *x*_{1}, *x*_{2}, … , *x*_{n} is any equation that can be written in the form

*z* = *a*_{0} + *a*_{1} *x*_{1} + *a*_{2} *x*_{2} + ··· + *a*_{n} *x*_{n}

In this form, we say that *z* is a linear function of *x*_{1}, *x*_{2}, … , *x*_{n}. The letters *a*_{0}, *a*_{1}, … , *a*_{n} are real numbers corresponding to constants.

Function notation applies to functions of several independent variables as well as functions of one independent variable. Recall that a linear function of one variable *x* named *f *would be written as *f (x)* = *a*_{0} + *a*_{1 }*x*. The independent variable for the function is placed in parentheses after the name to distinguish the variables from the constants. For a linear function of *n* independent variables, the *n* independent variables are placed in the parentheses after the name to give

*f* (*x*_{1}, *x*_{2}, … , *x*_{n}) = *a*_{0} + *a*_{1} *x*_{1} + *a*_{2} *x*_{2} + ··· + *a*_{n} *x*_{n}

**Example 1 Find Function Values**

If *f* (*x*_{1}, *x*_{2}, *x3*) = 10 – 2*x*_{1} + *x*_{2}+ 3*x*_{3}, find the value of *f *(6, -1, 2).

**Solution ** Substitute *x*_{1} = 6, *x*_{2} = -1, and *x*_{3} = 2 into the function to yield

*f* (6, -1, 2) = 10 – 2(6) + (-1)+ 3(2) = 3