To determine which of the given equations represents a direct variation between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex], let's first recall the definition of direct variation.
In direct variation, [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex]. Mathematically, this is expressed as:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is a constant.
Now, let's examine each option:
Option A: [tex]\( y = 9x \)[/tex]
- This equation fits the form of direct variation ([tex]\( y = kx \)[/tex]) where [tex]\( k = 9 \)[/tex].
- Therefore, this is an example of direct variation.
Option B: [tex]\( y = \frac{9}{x} \)[/tex]
- This equation represents an inverse variation where [tex]\( y \)[/tex] is inversely proportional to [tex]\( x \)[/tex].
- This does not fit the direct variation form [tex]\( y = kx \)[/tex].
Option C: [tex]\( y = \frac{x}{9} \)[/tex]
- This equation represents [tex]\( y \)[/tex] as a fraction of [tex]\( x \)[/tex] divided by the constant 9.
- This can be re-written as [tex]\( y = \frac{1}{9}x \)[/tex], which is a form of direct variation where [tex]\( k = \frac{1}{9} \)[/tex].
- Although this can still be considered a case of direct variation, for the purposes of this question, it may be more straightforward to recognize a simpler form like Option A.
Option D: [tex]\( y = x + 9 \)[/tex]
- This equation is a linear equation with a constant term.
- It does not fit the form [tex]\( y = kx \)[/tex] because of the added constant term (+9).
Among these options, the equation that clearly fits the definition of direct variation is:
[tex]\[ \boxed{A. \, y = 9x} \][/tex]
