9x + 3y = 12

Question

Consider the function represented by 9x + 3y = 12 with x as the independent variable. How can this function be written using function notation?

Answer 1

John, here is a good opportunity to introduce new terms:  first, the "Standard Form" of the Equation of a Line.  It looks like this:

      Ax + By = C

You line looks like that.

 

Although the equation does not yet identify the dependent and the independent variables, we usually let x be independent and y be dependent (as you mentioned).  This leads us to the "Slope-Intercept Form" of the Equation of a Line.  It looks like this:

       y = mx + b

 

And, there are others.

 

Since y depends on x, we are talking function here, so y=f(x).   We can write the slope-intercept form as:

     f(x) = mx + b

Note:  m is the slope of the line; b is the y-intercept (the value of y when x=0)

 

Starting with  9x+3y=12

   you should first divide all terms by 3, because   3x + y = 4  is precisely the same line.

Then, solve for y=f(x) = -3x + 4              (by subtracting 3x from both sides)

This is the slope-intercept form of the line.  The slope is -3 and the y-intercept is at the point (0,4).

Answer 2

First isolate the dependent variable, y.

 

9x + 3y = 12

3y = -9x + 12

y = -3x + 4

 

Replace the y with f(x)

 

f(x) = -3x + 4