The graphical representation of the equations ` x + 2y = 3 and 2x + 4 y + 7 = 0 ` gives a pair of

Options

A. parallel lines
B. intersecting lines
C. coincident lines
D. none of these

Answer

To determine the nature of the pair of lines represented by the equations \( x + 2y = 3 \) and \( 2x + 4y + 7 = 0 \), we will follow these steps:

### Step 1: Rewrite the equations in standard form
We need to express both equations in the form \( Ax + By + C = 0 \).

1. For the first equation:
\[ x + 2y = 3 \implies x + 2y - 3 = 0 \]
Here, \( A_1 = 1 \), \( B_1 = 2 \), and \( C_1 = -3 \).

2. For the second equation:
\[ 2x + 4y + 7 = 0 \]
Here, \( A_2 = 2 \), \( B_2 = 4 \), and \( C_2 = 7 \).

### Step 2: Calculate the ratios \( \frac{A_1}{A_2} \), \( \frac{B_1}{B_2} \), and \( \frac{C_1}{C_2} \)
Now, we will calculate the ratios of the coefficients:

1. Calculate \( \frac{A_1}{A_2} \):
\[ \frac{A_1}{A_2} = \frac{1}{2} \]

2. Calculate \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{2}{4} = \frac{1}{2} \]

3. Calculate \( \frac{C_1}{C_2} \):
\[ \frac{C_1}{C_2} = \frac{-3}{7} \]

### Step 3: Analyze the ratios
We have:
- \( \frac{A_1}{A_2} = \frac{1}{2} \)
- \( \frac{B_1}{B_2} = \frac{1}{2} \)
- \( \frac{C_1}{C_2} = \frac{-3}{7} \)

### Step 4: Determine the nature of the lines
According to the conditions for the nature of lines:
- If \( \frac{A_1}{A_2} = \frac{B_1}{B_2} \) but \( \frac{C_1}{C_2} \) is not equal to these ratios, the lines are **parallel**.
- If all three ratios are equal, the lines are **coincident**.
- If \( \frac{A_1}{A_2} \neq \frac{B_1}{B_2} \), the lines are **intersecting**.

In our case:
- \( \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{1}{2} \)
- \( \frac{C_1}{C_2} = \frac{-3}{7} \) (not equal to \( \frac{1}{2} \))

Thus, the lines are **parallel**.

### Final Answer
The graphical representation of the equations \( x + 2y = 3 \) and \( 2x + 4y + 7 = 0 \) gives a pair of **parallel lines**.

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