Let's analyze each of the given equations and determine their equivalence to the given expression [tex]\( 4s = t + 2 \)[/tex].
Given:
[tex]\[ 4s = t + 2 \][/tex]
1. Checking the equation [tex]\( s = t - 2 \)[/tex]
Substitute [tex]\( s \)[/tex] in the given equation:
[tex]\[ 4(t - 2) = t + 2 \][/tex]
Simplify:
[tex]\[ 4t - 8 = t + 2 \][/tex]
[tex]\[ 4t - t = 2 + 8 \][/tex]
[tex]\[ 3t = 10 \][/tex]
This simplifies to:
[tex]\[ t = \frac{10}{3} \][/tex]
Which means for a general case, it doesn't simplify to the given equation, hence not equivalent.
2. Checking the equation [tex]\( s = \frac{4}{t + 2} \)[/tex]
Substitute [tex]\( s \)[/tex] in the given equation:
[tex]\[ 4 \left(\frac{4}{t + 2}\right) = t + 2 \][/tex]
Simplify:
[tex]\[ \frac{16}{t + 2} = t + 2 \][/tex]
Multiply both sides by [tex]\( t + 2 \)[/tex] to clear the fraction:
[tex]\[ 16 = (t + 2)^2 \][/tex]
[tex]\[ 16 = t^2 + 4t + 4 \][/tex]
[tex]\[ t^2 + 4t + 4 - 16 = 0 \][/tex]
[tex]\[ t^2 + 4t - 12 = 0 \][/tex]
This is a quadratic equation and doesn't simplify to the given formula, hence not equivalent.
3. Checking the equation [tex]\( s = \frac{t + 2}{4} \)[/tex]
Substitute [tex]\( s \)[/tex] in the given equation:
[tex]\[ 4 \left(\frac{t + 2}{4}\right) = t + 2 \][/tex]
Simplify:
[tex]\[ t + 2 = t + 2 \][/tex]
This is a true statement, which means this equation is equivalent to the given equation.
4. Checking the equation [tex]\( s = t + 6 \)[/tex]
Substitute [tex]\( s \)[/tex] in the given equation:
[tex]\[ 4(t + 6) = t + 2 \][/tex]
Simplify:
[tex]\[ 4t + 24 = t + 2 \][/tex]
[tex]\[ 4t - t = 2 - 24 \][/tex]
[tex]\[ 3t = -22 \][/tex]
[tex]\[ t = -\frac{22}{3} \][/tex]
This doesn't match the given equation for a general [tex]\( t \)[/tex], hence not equivalent.
Based on this analysis:
[tex]\[ (s = \frac{t + 2}{4}) \][/tex] is the equation that is equivalent to [tex]\( 4s = t + 2 \)[/tex].
