Solve for x and y ,
8x + 5y - 11 = 0
3x - 4y - 10 = 0

Answer

To solve the system of equations:

1. **Write down the equations:**
\[ 8x + 5y - 11 = 0 \quad \text{(Equation 1)} \]
\[ 3x - 4y - 10 = 0 \quad \text{(Equation 2)} \]

2. **Rearrange both equations to standard form:**
\[ 8x + 5y = 11 \quad \text{(Equation 1)} \]
\[ 3x - 4y = 10 \quad \text{(Equation 2)} \]

3. **Find a common multiple for the coefficients of \(x\):**
The coefficients of \(x\) in the two equations are 8 and 3. The least common multiple (LCM) of 8 and 3 is 24.

4. **Multiply the equations to make the coefficients of \(x\) equal to 24:**
- Multiply Equation 1 by 3:
\[ 3(8x + 5y) = 3(11) \implies 24x + 15y = 33 \quad \text{(Equation 3)} \]
- Multiply Equation 2 by 8:
\[ 8(3x - 4y) = 8(10) \implies 24x - 32y = 80 \quad \text{(Equation 4)} \]

5. **Set up the new equations:**
\[ 24x + 15y = 33 \quad \text{(Equation 3)} \]
\[ 24x - 32y = 80 \quad \text{(Equation 4)} \]

6. **Subtract Equation 3 from Equation 4 to eliminate \(x\):**
\[ (24x - 32y) - (24x + 15y) = 80 - 33 \]
This simplifies to:
\[ -32y - 15y = 47 \implies -47y = 47 \]

7. **Solve for \(y\):**
\[ y = \frac{47}{-47} = -1 \]

8. **Substitute \(y\) back into one of the original equations to find \(x\):**
Using Equation 1:
\[ 8x + 5(-1) = 11 \]
This simplifies to:
\[ 8x - 5 = 11 \]
Adding 5 to both sides:
\[ 8x = 16 \]
Dividing by 8:
\[ x = 2 \]

9. **Final solution:**
\[ x = 2, \quad y = -1 \]