`3x-7y=12,kx+21y=-35`
For which of the following values of k will the system of equations above have no solution?

Options

A. 9
B. 3
C. `-3`
D. `-9`

Answer

To determine the value of \( k \) for which the system of equations has no solution, we need to analyze the given equations:

1. **Equations**:
- \( 3x - 7y = 12 \) (Equation 1)
- \( kx + 21y = -35 \) (Equation 2)

2. **Understanding No Solution**:
- A system of equations has no solution if the lines represented by the equations are parallel. This occurs when the slopes of the two lines are equal but the lines have different y-intercepts.

3. **Finding the Slopes**:
- The slope of a line in the form \( Ax + By = C \) can be calculated using the formula \( \text{slope} = -\frac{A}{B} \).

- For Equation 1:
\[ \text{slope of Equation 1} = -\frac{3}{-7} = \frac{3}{7} \]

- For Equation 2:
\[ \text{slope of Equation 2} = -\frac{k}{21} \]

4. **Setting the Slopes Equal**:
- For the lines to be parallel, we set the slopes equal to each other:
\[ \frac{3}{7} = -\frac{k}{21} \]

5. **Cross-Multiplying**:
- To eliminate the fractions, we cross-multiply:
\[ 3 \cdot 21 = -7k \]
\[ 63 = -7k \]

6. **Solving for \( k \)**:
- Now, we solve for \( k \):
\[ k = -\frac{63}{7} = -9 \]

7. **Conclusion**:
- The value of \( k \) for which the system of equations has no solution is \( k = -9 \).