If `x = (1 + 2y)/(2 + y) and y = ( 1 + 2t)/( 2 + t),` then x equals

Options

A. `(1 + 2t)/(3 + t)`
B. `(3 + 2t)/(2 + 3t)`
C. `(5t + 4)/(4 t + 5)`
D. `(5t + 6)/(6t + 5)`

Answer

To solve for \( x \) given the equations \( x = \frac{1 + 2y}{2 + y} \) and \( y = \frac{1 + 2t}{2 + t} \), we will substitute the expression for \( y \) into the equation for \( x \).

### Step-by-Step Solution:

1. **Substitute \( y \) in terms of \( t \)**:
\[ y = \frac{1 + 2t}{2 + t} \]
Now substitute this expression for \( y \) into the equation for \( x \):
\[ x = \frac{1 + 2\left(\frac{1 + 2t}{2 + t}\right)}{2 + \left(\frac{1 + 2t}{2 + t}\right)} \]

2. **Simplify the numerator**:
\[ x = \frac{1 + \frac{2(1 + 2t)}{2 + t}}{2 + \frac{1 + 2t}{2 + t}} \]
The numerator becomes:
\[ 1 + \frac{2(1 + 2t)}{2 + t} = \frac{(2 + t) + 2(1 + 2t)}{2 + t} = \frac{2 + t + 2 + 4t}{2 + t} = \frac{4 + 5t}{2 + t} \]

3. **Simplify the denominator**:
The denominator becomes:
\[ 2 + \frac{1 + 2t}{2 + t} = \frac{2(2 + t) + (1 + 2t)}{2 + t} = \frac{4 + 2t + 1 + 2t}{2 + t} = \frac{5 + 4t}{2 + t} \]

4. **Combine the results**:
Now we can rewrite \( x \):
\[ x = \frac{\frac{4 + 5t}{2 + t}}{\frac{5 + 4t}{2 + t}} \]
Since the denominators are the same, they cancel out:
\[ x = \frac{4 + 5t}{5 + 4t} \]

### Final Result:
Thus, the value of \( x \) is:
\[ x = \frac{4 + 5t}{5 + 4t} \]