Which of the following system of equations has no solution?

Options

A. `3x-y=2, 9x-3y=6`
B. `4x-7y+28=0, 5y-7x+9=0`
C. `3x-5y-11=0, 6x-10y-7=0`
D. None of these

Answer

To determine which system of equations has no solution, we need to analyze the given options based on the conditions for linear equations. The key is to identify when two lines represented by the equations are parallel, which means they will never intersect and thus have no solutions.

### Step-by-Step Solution:

1. **Understand the Conditions for Solutions**:
- For two linear equations in the form:
\[ a_1x + b_1y + c_1 = 0 \]
\[ a_2x + b_2y + c_2 = 0 \]
- The conditions are:
- If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), there is one unique solution (the lines intersect).
- If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), there are no solutions (the lines are parallel).
- If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), there are infinite solutions (the lines coincide).

2. **Analyze Each Option**:
- **Option 1**:
- Equations: \(3x - y - 2 = 0\) and \(9x - 3y - 6 = 0\)
- Coefficients: \(a_1 = 3, b_1 = -1, c_1 = -2\) and \(a_2 = 9, b_2 = -3, c_2 = -6\)
- Calculate ratios:
\[ \frac{a_1}{a_2} = \frac{3}{9} = \frac{1}{3}, \quad \frac{b_1}{b_2} = \frac{-1}{-3} = \frac{1}{3}, \quad \frac{c_1}{c_2} = \frac{-2}{-6} = \frac{1}{3} \]
- Since all ratios are equal, this system has infinite solutions. **Not the answer**.

- **Option 2**:
- Equations: \(4x - 7y + 28 = 0\) and \(5x - 7y + 9 = 0\)
- Coefficients: \(a_1 = 4, b_1 = -7, c_1 = 28\) and \(a_2 = 5, b_2 = -7, c_2 = 9\)
- Calculate ratios:
\[ \frac{a_1}{a_2} = \frac{4}{5}, \quad \frac{b_1}{b_2} = \frac{-7}{-7} = 1 \]
- Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), this system has a unique solution. **Not the answer**.

- **Option 3**:
- Equations: \(3x - 5y - 11 = 0\) and \(6x - 10y + 7 = 0\)
- Coefficients: \(a_1 = 3, b_1 = -5, c_1 = -11\) and \(a_2 = 6, b_2 = -10, c_2 = 7\)
- Calculate ratios:
\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-5}{-10} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{-11}{7} \]
- Here, \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) but \(\frac{c_1}{c_2} \neq \frac{1}{2}\). Thus, this system has no solutions (the lines are parallel). **This is the answer**.

### Conclusion:
The system of equations that has no solution is **Option 3**.