Find the value of `x` and `y` in given equations `x+2y+3=0 and 4x-5y+1=0` .

Answer

To find the values of \( x \) and \( y \) in the given equations:

1. **Write the equations:**
\[ x + 2y + 3 = 0 \quad \text{(Equation 1)} \]
\[ 4x - 5y + 1 = 0 \quad \text{(Equation 2)} \]

2. **Rearrange Equation 1:**
\[ x + 2y = -3 \quad \text{(Rearranging Equation 1)} \]

3. **Rearrange Equation 2:**
\[ 4x - 5y = -1 \quad \text{(Rearranging Equation 2)} \]

4. **Multiply Equation 1 by 4:**
\[ 4(x + 2y) = 4(-3) \]
This gives us:
\[ 4x + 8y = -12 \quad \text{(Equation 3)} \]

5. **Now we have two equations:**
\[ 4x + 8y = -12 \quad \text{(Equation 3)} \]
\[ 4x - 5y = -1 \quad \text{(Equation 2)} \]

6. **Subtract Equation 2 from Equation 3:**
\[ (4x + 8y) - (4x - 5y) = -12 - (-1) \]
Simplifying this gives:
\[ 8y + 5y = -12 + 1 \]
\[ 13y = -11 \]

7. **Solve for \( y \):**
\[ y = \frac{-11}{13} \]

8. **Substitute the value of \( y \) back into Equation 1:**
\[ x + 2\left(\frac{-11}{13}\right) = -3 \]
This simplifies to:
\[ x - \frac{22}{13} = -3 \]

9. **Rearranging gives:**
\[ x = -3 + \frac{22}{13} \]
To combine these, convert \(-3\) to a fraction:
\[ -3 = \frac{-39}{13} \]
Thus:
\[ x = \frac{-39 + 22}{13} = \frac{-17}{13} \]

10. **Final values:**
\[ x = \frac{-17}{13}, \quad y = \frac{-11}{13} \]