Sure! Let's go through the problem step by step to solve for [tex]\( t \)[/tex] in the given equation:
[tex]\[
-\frac{3}{t} - 9 = s + 5
\][/tex]
### Step 1: Isolate the term involving [tex]\( t \)[/tex]
First, add 9 to both sides of the equation to isolate the fraction involving [tex]\( t \)[/tex]:
[tex]\[
-\frac{3}{t} = s + 5 + 9
\][/tex]
Simplify the right-hand side of the equation:
[tex]\[
-\frac{3}{t} = s + 14
\][/tex]
### Step 2: Solve for [tex]\( \frac{1}{t} \)[/tex]
Now, we need to isolate the [tex]\( \frac{3}{t} \)[/tex] term. To do this, we will divide both sides of the equation by -1:
[tex]\[
\frac{3}{t} = -(s + 14)
\][/tex]
### Step 3: Invert both sides to solve for [tex]\( t \)[/tex]
To solve for [tex]\( t \)[/tex], we take the reciprocal of both sides of the equation:
[tex]\[
t = \frac{3}{-(s + 14)}
\][/tex]
### Step 4: Simplify the expression
Simplifying the right-hand side, we get:
[tex]\[
t = -\frac{3}{s + 14}
\][/tex]
So the correct solution is:
[tex]\[
t = -\frac{3}{s + 14}
\][/tex]
### Verification of Given Options
Let's verify the options provided:
1. [tex]\( t = -\frac{3}{s + 14} \)[/tex] [tex]\(\rightarrow\)[/tex] This matches our result.
2. [tex]\( t = \frac{14}{s} \)[/tex]
3. [tex]\( t = -\frac{s + 14}{3} \)[/tex]
4. [tex]\( t = \frac{s + 14}{-4} \)[/tex]
Clearly, option 1, [tex]\( t = -\frac{3}{s + 14} \)[/tex], is the correct solution.
