Find the value of x and y from the given pair of linear equations:
` 2x - 3y + 4 =0 , x + 2y - 5 =0 `

Answer

To solve the given pair of linear equations:

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

2. **Rearranging the Equations**:
- For Equation 1, we can rearrange it to isolate the constant on the right:
\[ 2x - 3y = -4 \quad \text{(Equation 1)} \]
- For Equation 2, we can also rearrange it:
\[ x + 2y = 5 \quad \text{(Equation 2)} \]

3. **Using the Elimination Method**:
- We will eliminate one variable by making the coefficients of \(x\) the same in both equations. To do this, we can multiply Equation 2 by 2:
\[ 2(x + 2y) = 2(5) \]
This gives us:
\[ 2x + 4y = 10 \quad \text{(Modified Equation 2)} \]

4. **Setting the Equations for Elimination**:
- Now we have:
\[ 2x - 3y = -4 \quad \text{(Equation 1)} \]
\[ 2x + 4y = 10 \quad \text{(Modified Equation 2)} \]

5. **Subtracting the Equations**:
- We can subtract Equation 1 from the Modified Equation 2:
\[ (2x + 4y) - (2x - 3y) = 10 - (-4) \]
Simplifying this gives:
\[ 2x + 4y - 2x + 3y = 10 + 4 \]
\[ 7y = 14 \]

6. **Solving for \(y\)**:
- Now, divide both sides by 7:
\[ y = \frac{14}{7} = 2 \]

7. **Finding the Value of \(x\)**:
- Substitute \(y = 2\) back into one of the original equations. We will use Equation 2:
\[ x + 2(2) = 5 \]
\[ x + 4 = 5 \]
\[ x = 5 - 4 = 1 \]

8. **Final Values**:
- The solution to the equations is:
\[ x = 1, \quad y = 2 \]