What should be added to `5 (2x -y)` to obtain `4 (2x -3y )+ 5 (x + y)` ?

Options

A. `3x - 14 y`
B. `3x + 13 y`
C. `3 x - 2y`
D. `13 x + 3y`

Answer

To solve the problem, we need to find what should be added to \( 5(2x - y) \) to obtain \( 4(2x - 3y) + 5(x + y) \).

### Step-by-step Solution:

1. **Expand both sides of the equation:**
- Start with the left side:
\[ 5(2x - y) = 10x - 5y \]
- Now expand the right side:
\[ 4(2x - 3y) + 5(x + y) = 8x - 12y + 5x + 5y \]
Combine like terms:
\[ 8x + 5x - 12y + 5y = 13x - 7y \]

2. **Set up the equation:**
- We need to find \( m \) such that:
\[ 10x - 5y + m = 13x - 7y \]

3. **Isolate \( m \):**
- Rearranging gives:
\[ m = (13x - 7y) - (10x - 5y) \]
- Simplify the right side:
\[ m = 13x - 7y - 10x + 5y \]
Combine like terms:
\[ m = (13x - 10x) + (-7y + 5y) = 3x - 2y \]

4. **Conclusion:**
- Therefore, the value that should be added is:
\[ m = 3x - 2y \]