Which of the folowing pair represent equation of parallel straight lines.

Options

A. `2x + 3y = 4, 4x + 6y = 9`
B. `x + 2y = 4, 2x + y = 4`
C. `y = 3x + 5, x = 3y +5`
D. None of these

Answer

To determine which pair of equations represents parallel straight lines, we will use the condition for parallel lines in the form of linear equations. The condition states that for two lines represented by the equations \(A_1x + B_1y = C_1\) and \(A_2x + B_2y = C_2\), the lines are parallel if:

\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2} \]

Now, let's analyze each option step-by-step.

### Step 1: Analyze the First Option
The first option is:
1. \(2x + 3y = 4\) (Equation 1)
2. \(4x + 6y = 9\) (Equation 2)

**Identify coefficients:**
- For Equation 1: \(A_1 = 2\), \(B_1 = 3\), \(C_1 = 4\)
- For Equation 2: \(A_2 = 4\), \(B_2 = 6\), \(C_2 = 9\)

**Calculate ratios:**
- \(\frac{A_1}{A_2} = \frac{2}{4} = \frac{1}{2}\)
- \(\frac{B_1}{B_2} = \frac{3}{6} = \frac{1}{2}\)
- \(\frac{C_1}{C_2} = \frac{4}{9}\)

**Check conditions:**
- Since \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\) and \(\frac{C_1}{C_2} \neq \frac{1}{2}\), the first option satisfies the condition for parallel lines.

### Step 2: Analyze the Second Option
The second option is:
1. \(x + 2y = 4\) (Equation 1)
2. \(2x + y = 4\) (Equation 2)

**Identify coefficients:**
- For Equation 1: \(A_1 = 1\), \(B_1 = 2\), \(C_1 = 4\)
- For Equation 2: \(A_2 = 2\), \(B_2 = 1\), \(C_2 = 4\)

**Calculate ratios:**
- \(\frac{A_1}{A_2} = \frac{1}{2}\)
- \(\frac{B_1}{B_2} = \frac{2}{1} = 2\)

**Check conditions:**
- Since \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\), the second option does not satisfy the condition for parallel lines.

### Step 3: Analyze the Third Option
The third option is:
1. \(y = 3x + 5\) (Equation 1)
2. \(x = 3y + 5\) (Equation 2)

**Rearranging Equation 2:**
- Rearranging \(x = 3y + 5\) gives us \(-3y + x = 5\) or \(x - 3y = 5\).

**Identify coefficients:**
- For Equation 1: \(A_1 = -3\), \(B_1 = 1\), \(C_1 = 5\)
- For Equation 2: \(A_2 = 1\), \(B_2 = -3\), \(C_2 = 5\)

**Calculate ratios:**
- \(\frac{A_1}{A_2} = \frac{-3}{1} = -3\)
- \(\frac{B_1}{B_2} = \frac{1}{-3}\)

**Check conditions:**
- Since \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\), the third option does not satisfy the condition for parallel lines.

### Final Conclusion
Only the first option satisfies the condition for parallel lines. Therefore, the answer is:

**Answer: The first option \(2x + 3y = 4\) and \(4x + 6y = 9\) represents parallel straight lines.**

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