If `3s+5t=10`, and `2s-t=7`, what is the value of `(1)/(2^(s))+3t` ?

Options

A. 0
B. `1.5`
C. `2.8`
D. `3.4`

Explanation

To solve the problem step by step, we will first work with the two equations provided:

1. **Given Equations**:
- \( 3s + 5t = 10 \) (Equation 1)
- \( 2s - t = 7 \) (Equation 2)

2. **Rearranging Equation 2**:
We can rearrange Equation 2 to express \( t \) in terms of \( s \):
\[ t = 2s - 7 \]

3. **Substituting \( t \) into Equation 1**:
Now, substitute \( t \) from Equation 2 into Equation 1:
\[ 3s + 5(2s - 7) = 10 \]
Simplifying this gives:
\[ 3s + 10s - 35 = 10 \]
Combine like terms:
\[ 13s - 35 = 10 \]

4. **Solving for \( s \)**:
Add 35 to both sides:
\[ 13s = 45 \]
Now, divide by 13:
\[ s = \frac{45}{13} \]

5. **Finding \( t \)**:
Substitute \( s \) back into the rearranged Equation 2 to find \( t \):
\[ t = 2\left(\frac{45}{13}\right) - 7 \]
This simplifies to:
\[ t = \frac{90}{13} - \frac{91}{13} = -\frac{1}{13} \]

6. **Calculating the Expression**:
Now we need to find the value of \( \frac{1}{2^s} + 3t \):
\[ \frac{1}{2^{\frac{45}{13}}} + 3\left(-\frac{1}{13}\right) \]
Simplifying \( 3t \):
\[ 3t = -\frac{3}{13} \]
Therefore, the expression becomes:
\[ \frac{1}{2^{\frac{45}{13}}} - \frac{3}{13} \]

7. **Final Value**:
The final value of the expression is:
\[ \frac{1}{2^{\frac{45}{13}}} - \frac{3}{13} \]