Given the linear equation `3x - 2y + 7 =0,` write another linear equation in two variables such that the geometrical representation of the pair so formed is
(i)intersecting lines
(ii) parallel lines
(iii) coincident lines

Answer

To solve the problem, we need to find three different linear equations based on the given equation \(3x - 2y + 7 = 0\) such that:

1. The lines represented by the equations intersect.
2. The lines represented by the equations are parallel.
3. The lines represented by the equations are coincident.

### Step 1: Intersecting Lines
To find an equation for intersecting lines, we need to ensure that the ratio of the coefficients of \(x\) and \(y\) from the two equations are not equal.

Given the first equation:
\[ 3x - 2y + 7 = 0 \]
Here, \(a_1 = 3\), \(b_1 = -2\), and \(c_1 = 7\).

Let’s choose:
- \(a_2 = 4\)
- \(b_2 = 1\)
- \(c_2 = -1\)

Now, we check the condition:
\[ \frac{a_1}{a_2} = \frac{3}{4}, \quad \frac{b_1}{b_2} = \frac{-2}{1} \]
Since \(\frac{3}{4} \neq \frac{-2}{1}\), the lines will intersect.

Thus, the second equation is:
\[ 4x + y - 1 = 0 \]

### Step 2: Parallel Lines
For parallel lines, the ratio of the coefficients of \(x\) and \(y\) must be equal, but the ratio of the constant terms must not be equal.

Using the same coefficients from the first equation:
- \(a_1 = 3\), \(b_1 = -2\), \(c_1 = 7\)

Let’s choose:
- \(a_2 = 6\)
- \(b_2 = -4\)
- \(c_2 = 3\)

Now, we check the condition:
\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-2}{-4} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{7}{3} \]
Since \(\frac{1}{2} = \frac{1}{2}\) and \(\frac{7}{3} \neq \frac{1}{2}\), the lines are parallel.

Thus, the second equation is:
\[ 6x - 4y + 3 = 0 \]

### Step 3: Coincident Lines
For coincident lines, the ratios of all coefficients must be equal.

Using the same coefficients from the first equation:
- \(a_1 = 3\), \(b_1 = -2\), \(c_1 = 7\)

Let’s choose:
- \(a_2 = 6\)
- \(b_2 = -4\)
- \(c_2 = 14\)

Now, we check the condition:
\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-2}{-4} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2} \]
Since all ratios are equal, the lines are coincident.

Thus, the second equation is:
\[ 6x - 4y + 14 = 0 \]

### Summary of the Equations
1. **Intersecting Lines**: \(4x + y - 1 = 0\)
2. **Parallel Lines**: \(6x - 4y + 3 = 0\)
3. **Coincident Lines**: \(6x - 4y + 14 = 0\)