To determine which equation represents inverse variation between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we recall that in an inverse variation, one variable is equal to a constant divided by the other variable. Mathematically, we express inverse variation as:
[tex]\[ y = \frac{k}{x} \][/tex]
where [tex]\( k \)[/tex] is a constant.
Let's examine each of the given options:
A. [tex]\( y = 6x \)[/tex]
- This is a direct variation. When [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases proportionally, as they are directly multiplied by a constant (6 in this case). Hence, this is not an inverse variation.
B. [tex]\( y = x + 6 \)[/tex]
- This is neither direct nor inverse variation. It represents a linear relationship with a slope of 1 and a y-intercept of 6. The variables are not inversely related here.
C. [tex]\( y = \frac{x}{6} \)[/tex]
- This is also a direct variation with a constant of [tex]\(\frac{1}{6}\)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases proportionally by [tex]\(\frac{x}{6}\)[/tex]. This does not represent inverse variation.
D. [tex]\( y = \frac{6}{x} \)[/tex]
- This equation fits the definition of inverse variation. Here, [tex]\( y \)[/tex] is equal to a constant (6) divided by [tex]\( x \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] decreases in such a way that their product [tex]\( xy \)[/tex] remains a constant.
Thus, the equation that represents an inverse variation between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ \boxed{y = \frac{6}{x}} \][/tex]
So, the correct answer is:
[tex]\[ \text{Option D. } y = \frac{6}{x} \][/tex]
