Sure, let's solve the given equation step by step to determine which equation correctly solves for [tex]\( u \)[/tex].
We start with the equation:
[tex]\[ 7 = \frac{1}{2} a t^2 + u t \][/tex]
Our goal is to solve for [tex]\( u \)[/tex].
1. Step 1: Isolate the term involving [tex]\( u \)[/tex] on one side
Subtract [tex]\(\frac{1}{2} a t^2\)[/tex] from both sides to move it away from [tex]\( u t \)[/tex]:
[tex]\[ 7 - \frac{1}{2} a t^2 = u t \][/tex]
2. Step 2: Solve for [tex]\( u \)[/tex]
To isolate [tex]\( u \)[/tex], divide both sides of the equation by [tex]\( t \)[/tex]:
[tex]\[ u = \frac{7 - \frac{1}{2} a t^2}{t} \][/tex]
So, the equation that correctly solves for [tex]\( u \)[/tex] is:
[tex]\[ u = \frac{7 - \frac{1}{2} a t^2}{t} \][/tex]
Now, let's compare this solution to the given options:
1. [tex]\( u = \frac{14 - a t^2}{t} \)[/tex]
2. [tex]\( u = 7 - \frac{1}{2} a t^2 - t \)[/tex]
3. [tex]\( u = \frac{7 - \frac{1}{2} a t^2}{t} \)[/tex]
4. [tex]\( u = 14 - a t^2 - 2 t \)[/tex]
Clearly, option (3) matches our derived solution. Therefore, the correct equation that solves for [tex]\( u \)[/tex] is:
[tex]\[ u = \frac{7 - \frac{1}{2} a t^2}{t} \][/tex]
