To determine which equation represents a proportional relationship with a constant of proportionality equal to -10, we need to understand the form of such a relationship. In mathematics, a proportional relationship can be described by an equation of the form:
[tex]\[ y = kx \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality. Here, we require [tex]\( k \)[/tex] to be -10. This means our equation should look like:
[tex]\[ y = -10x \][/tex]
Now, let's match this structure against the given options:
1. [tex]\( y = x - 10 \)[/tex]
This equation is not in the form [tex]\( y = kx \)[/tex] because there is a subtraction operation, not a multiplication by a constant. It represents a linear relationship but not a proportional one with the required constant.
2. [tex]\( y = \frac{x}{-10} \)[/tex]
This equation can be rewritten as [tex]\( y = \frac{1}{-10} x \)[/tex]. Here, the constant of proportionality is [tex]\(\frac{1}{-10}\)[/tex], not -10, so it does not satisfy our condition.
3. [tex]\( y = -10x \)[/tex]
This equation is in the form [tex]\( y = kx \)[/tex] with [tex]\( k = -10 \)[/tex]. Hence, it correctly represents a proportional relationship with the constant of proportionality equal to -10.
4. [tex]\( y = -10 \)[/tex]
This equation describes a horizontal line where [tex]\( y \)[/tex] is constantly -10, regardless of [tex]\( x \)[/tex]. It does not fit the form [tex]\( y = kx \)[/tex].
Thus, the correct equation that represents a proportional relationship with a constant of proportionality equal to -10 is:
[tex]\[
\boxed{3}
\][/tex]
