Which one of the following statement is correct ? The system of linear equations, `2x + 3y = 4 and 4x + 6y = 7`, has

Options

A. no solution
B. a unique solution
C. exactly 3 solutions
D. an infinite number of solutions

Explanation

To determine the nature of the system of linear equations given by:

1. \( 2x + 3y = 4 \)
2. \( 4x + 6y = 7 \)

we can analyze the coefficients and constants of the equations.

### Step 1: Identify coefficients and constants
From the equations, we can identify:
- For the first equation \( (a_1, b_1, c_1) = (2, 3, 4) \)
- For the second equation \( (a_2, b_2, c_2) = (4, 6, 7) \)

### Step 2: Check the ratio of coefficients
Now, we will check the ratio of the coefficients:
\[ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \]
\[ \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \]

### Step 3: Check the ratio of constants
Next, we check the ratio of the constants:
\[ \frac{c_1}{c_2} = \frac{4}{7} \]

### Step 4: Analyze the results
We have:
\[ \frac{a_1}{a_2} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{1}{2}, \quad \text{and} \quad \frac{c_1}{c_2} = \frac{4}{7} \]

Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) but \(\frac{c_1}{c_2}\) is not equal to these ratios, we can conclude that the system of equations does not have a solution. This is a case of parallel lines that do not intersect.

### Conclusion
Thus, the correct statement is that the system of linear equations has **no solution**.

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