To determine which equation can be used to solve for acceleration, let's analyze each given equation one-by-one:
1. [tex]\( t = \frac{\Delta v}{a} \)[/tex]: This equation expresses time [tex]\( t \)[/tex] in terms of the change in velocity [tex]\( \Delta v \)[/tex] and acceleration [tex]\( a \)[/tex]. To solve for acceleration [tex]\( a \)[/tex] from this equation, we would rearrange it to:
[tex]\[
a = \frac{\Delta v}{t}
\][/tex]
While this rearranged form is correct for acceleration, the original format does not explicitly solve for [tex]\( a \)[/tex].
2. [tex]\( v_f = at + v \)[/tex]: This equation relates the final velocity [tex]\( v_f \)[/tex], initial velocity [tex]\( v \)[/tex], acceleration [tex]\( a \)[/tex], and time [tex]\( t \)[/tex]. Rearranging this equation to solve for acceleration [tex]\( a \)[/tex] yields:
[tex]\[
a = \frac{v_f - v}{t}
\][/tex]
This is a correct and direct way to express acceleration in terms of known variables.
3. [tex]\( a = \frac{d}{t} \)[/tex]: This equation suggests that acceleration is the distance [tex]\( d \)[/tex] divided by time [tex]\( t \)[/tex]. This is incorrect because acceleration is actually the change in velocity over time, not distance over time.
4. [tex]\( \Delta v = \frac{a}{t} \)[/tex]: This equation implies that the change in velocity [tex]\( \Delta v \)[/tex] is equal to acceleration [tex]\( a \)[/tex] divided by time [tex]\( t \)[/tex], which is incorrect in terms of properties of physical quantities. The correct relationship should involve multiplication, not division.
Among these equations, the one that directly and correctly solves for acceleration [tex]\( a \)[/tex] is:
[tex]\[
v_f = a t + v \quad \Rightarrow \quad a = \frac{v_f - v}{t}
\][/tex]
Therefore, the correct equation that can be used to solve for acceleration is given by:
[tex]\[
\boxed{1}
\][/tex]
