Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
[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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.