To determine which of the given equations exemplifies an inverse variation between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we need to recall the definition of inverse variation. In inverse variation, the product of the two variables is equal to a constant. Mathematically, this relationship can be expressed as:
[tex]\[ y = \frac{k}{x} \][/tex]
where [tex]\( k \)[/tex] is a constant.
Let's evaluate each of the given options one-by-one:
Option A: [tex]\( y = \frac{x}{9} \)[/tex]
- This equation represents a direct variation rather than an inverse variation. Here, [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex], but scaled by the factor [tex]\(\frac{1}{9}\)[/tex].
Option B: [tex]\( y = 9x \)[/tex]
- This equation also represents a direct variation. In this case, [tex]\( y \)[/tex] is directly proportional to [tex]\( x \)[/tex] with a proportionality factor of 9.
Option C: [tex]\( y = \frac{9}{x} \)[/tex]
- This equation fits the form [tex]\( y = \frac{k}{x} \)[/tex], where [tex]\( k = 9 \)[/tex]. It describes an inverse variation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Here, the product [tex]\( xy = 9 \)[/tex], which is a constant.
Option D: [tex]\( y = x + 9 \)[/tex]
- This equation represents a linear relationship where [tex]\( y \)[/tex] is shifted by 9 units from [tex]\( x \)[/tex]. It is neither direct nor inverse variation.
Among the given options, option C: [tex]\( y = \frac{9}{x} \)[/tex] is the equation that illustrates inverse variation because it adheres to the form [tex]\( y = \frac{k}{x} \)[/tex].
Thus, the correct answer is:
Option C: [tex]\( y = \frac{9}{x} \)[/tex].
