Given the linear equation `2x + 3y – 12 = 0`, write another linear equation in these variables, such that. geometrical representation of the pair so formed is
(i) Parallel Lines (ii) Coincident Lines

Answer

To solve the question, we need to write two different linear equations based on the given equation \(2x + 3y - 12 = 0\). We will find one equation that represents parallel lines and another that represents coincident lines.

### Step 1: Identify coefficients from the given equation
The given equation is:
\[ 2x + 3y - 12 = 0 \]
From this, we can identify:
- \(A_1 = 2\)
- \(B_1 = 3\)
- \(C_1 = -12\)

### Step 2: Find an equation for parallel lines
For two lines to be parallel, the condition is:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \quad \text{and} \quad \frac{C_1}{C_2} \neq \frac{A_1}{A_2} \]
Let's choose:
- \(A_2 = 4\)
- \(B_2 = 6\)

Now, we check the ratios:
\[ \frac{A_1}{A_2} = \frac{2}{4} = \frac{1}{2}, \quad \frac{B_1}{B_2} = \frac{3}{6} = \frac{1}{2} \]
These ratios are equal, which is correct for parallel lines. Now, we need to choose \(C_2\) such that:
\[ C_2 \neq -24 \quad (\text{since } C_1 = -12 \text{ and } \frac{C_1}{C_2} \text{ should not equal } \frac{1}{2}) \]
Let's take \(C_2 = 5\).

Now, we can write the equation:
\[ 4x + 6y + 5 = 0 \]

### Step 3: Find an equation for coincident lines
For two lines to be coincident, the condition is:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]
Let's choose:
- \(A_2 = 6\)
- \(B_2 = 9\)
- \(C_2 = -18\)

Now, we check the ratios:
\[ \frac{A_1}{A_2} = \frac{2}{6} = \frac{1}{3}, \quad \frac{B_1}{B_2} = \frac{3}{9} = \frac{1}{3}, \quad \frac{C_1}{C_2} = \frac{-12}{-18} = \frac{2}{3} \]
To make them equal, we can take \(C_2 = -18\) (since it should maintain the same ratio).

Now, we can write the equation:
\[ 6x + 9y - 18 = 0 \]

### Final Answers
1. **Equation for Parallel Lines**:
\[ 4x + 6y + 5 = 0 \]
2. **Equation for Coincident Lines**:
\[ 6x + 9y - 18 = 0 \]