To determine which equations are equivalent to \( d = r \cdot t \), let's solve the equation for both \( r \) and \( t \) step-by-step.
1. Solving for \( r \):
The original equation is:
[tex]\[
d = r \cdot t
\][/tex]
To solve for \( r \), divide both sides of the equation by \( t \):
[tex]\[
r = \frac{d}{t}
\][/tex]
2. Solving for \( t \):
The original equation is:
[tex]\[
d = r \cdot t
\][/tex]
To solve for \( t \), divide both sides of the equation by \( r \):
[tex]\[
t = \frac{d}{r}
\][/tex]
Now, let's match these manipulated forms of the equation to the given options:
1. \( r = \frac{d}{t} \)
2. \( r = d \cdot t \)
3. \( t = \frac{r}{d} \)
4. \( t = d \cdot r \)
Based on our derived equations \( r = \frac{d}{t} \) and \( t = \frac{d}{r} \):
- The equivalent form where \( r \) is expressed in terms of \( d \) and \( t \) is: \( r = \frac{d}{t} \) (Option 1)
- The equivalent form where \( t \) is expressed in terms of \( d \) and \( r \) is: \( t = \frac{d}{r} \) (Option 3)
Thus, the correct answers are options:
[tex]\( \boxed{1 \text{ and } 3} \)[/tex]
