Question 31 of 42

Which of the following equations is an example of direct variation between variables [tex]$x$[/tex] and [tex]$y$[/tex]?

A. [tex][tex]$y=9x$[/tex][/tex]

B. [tex]$y=\frac{9}{x}$[/tex]

C. [tex]$y=\frac{x}{9}$[/tex]

D. [tex][tex]$y=x+9$[/tex][/tex]



Answer :

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]