Let's determine whether each equation represents an inverse variation or a direct variation.
### Equation 1: [tex]\( y = 5x \)[/tex]
This equation can be rewritten in the form [tex]\( y = kx \)[/tex] where [tex]\( k \)[/tex] is a constant. In this case, [tex]\( k = 5 \)[/tex]. Equations of this form [tex]\( y = kx \)[/tex] represent direct variation because the product of [tex]\( x \)[/tex] and a constant results directly in [tex]\( y \)[/tex].
Conclusion: [tex]\( y = 5x \)[/tex] is a direct variation.
### Equation 2: [tex]\( ab = 5 \)[/tex]
This equation can be rewritten in the form [tex]\( xy = k \)[/tex] where [tex]\( k \)[/tex] is a constant. Here, [tex]\( k = 5 \)[/tex]. When two variables multiply to give a constant ([tex]\( k \)[/tex]), the relationship between them is called inverse variation. This means that when one variable increases, the other decreases so that their product remains constant.
Conclusion: [tex]\( ab = 5 \)[/tex] is an inverse variation.
### Equation 3: [tex]\( xy = -1 \)[/tex]
Similarly, this equation can also be rewritten in the form [tex]\( xy = k \)[/tex] where [tex]\( k \)[/tex] is a constant. In this case, [tex]\( k = -1 \)[/tex]. The form [tex]\( xy = k \)[/tex] indicates inverse variation because the product of the two variables is a constant, meaning that as one variable increases, the other must decrease to keep the product consistent.
Conclusion: [tex]\( xy = -1 \)[/tex] is an inverse variation.
To summarize, the determinations are as follows:
1. [tex]\( y = 5x \)[/tex] is a direct variation.
2. [tex]\( ab = 5 \)[/tex] is an inverse variation.
3. [tex]\( xy = -1 \)[/tex] is an inverse variation.
