To determine which equation is equivalent to [tex]\( 4s = t + 2 \)[/tex], we need to manipulate each given option and check if it can be transformed to match this equation. Let's go through each option step-by-step.
1. Option 1: [tex]\( s = t - 2 \)[/tex]
Substituting [tex]\( s = t - 2 \)[/tex] into the original equation [tex]\( 4s \)[/tex]:
[tex]\[
4s = 4(t - 2) = 4t - 8
\][/tex]
This simplifies to [tex]\( 4t - 8 \)[/tex], which is not equal to [tex]\( t + 2 \)[/tex]. Thus, this option is not equivalent.
2. Option 2: [tex]\( s = \frac{4}{t + 2} \)[/tex]
Substituting [tex]\( s = \frac{4}{t + 2} \)[/tex] into the original equation [tex]\( 4s \)[/tex]:
[tex]\[
4s = 4\left( \frac{4}{t + 2} \right) = \frac{16}{t + 2}
\][/tex]
This simplifies to [tex]\( \frac{16}{t + 2} \)[/tex], which is not equal to [tex]\( t + 2 \)[/tex]. Thus, this option is also not equivalent.
3. Option 3: [tex]\( s = \frac{t + 2}{4} \)[/tex]
Substituting [tex]\( s = \frac{t + 2}{4} \)[/tex] into the original equation [tex]\( 4s \)[/tex]:
[tex]\[
4s = 4\left( \frac{t + 2}{4} \right) = t + 2
\][/tex]
This simplifies to [tex]\( t + 2 \)[/tex], which matches our original equation. Thus, this option is equivalent.
4. Option 4: [tex]\( s = t + 6 \)[/tex]
Substituting [tex]\( s = t + 6 \)[/tex] into the original equation [tex]\( 4s \)[/tex]:
[tex]\[
4s = 4(t + 6) = 4t + 24
\][/tex]
This simplifies to [tex]\( 4t + 24 \)[/tex], which is not equal to [tex]\( t + 2 \)[/tex]. Thus, this option is not equivalent.
Based on these steps, the correct option equivalent to [tex]\( 4s = t + 2 \)[/tex] is:
Option 3: [tex]\( s = \frac{t + 2}{4} \)[/tex].
