Solve the following pair of linear equations.
`(x)/(2) + (2y)/(3)= -1 and x- (y)/(3)=3`

Answer

To solve the given pair of linear equations:

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

2. **Eliminate Fractions**:
To eliminate the fractions in equation (1), we can multiply through by 6 (the least common multiple of the denominators 2 and 3):
\[ 6 \left(\frac{x}{2}\right) + 6 \left(\frac{2y}{3}\right) = 6(-1) \]
This simplifies to:
\[ 3x + 4y = -6 \quad \text{(3)} \]

For equation (2), we can multiply through by 3 to eliminate the fraction:
\[ 3 \left(x - \frac{y}{3}\right) = 3(3) \]
This simplifies to:
\[ 3x - y = 9 \quad \text{(4)} \]

3. **Now we have a new system of equations**:
\[ 3x + 4y = -6 \quad \text{(3)} \]
\[ 3x - y = 9 \quad \text{(4)} \]

4. **Subtract equation (4) from equation (3)**:
\[ (3x + 4y) - (3x - y) = -6 - 9 \]
This simplifies to:
\[ 4y + y = -15 \]
\[ 5y = -15 \]

5. **Solve for \( y \)**:
\[ y = \frac{-15}{5} = -3 \]

6. **Substitute \( y \) back into equation (4) to find \( x \)**:
\[ 3x - (-3) = 9 \]
This simplifies to:
\[ 3x + 3 = 9 \]
\[ 3x = 9 - 3 \]
\[ 3x = 6 \]
\[ x = \frac{6}{3} = 2 \]

7. **Final Solution**:
The solution to the pair of linear equations is:
\[ x = 2, \quad y = -3 \]