One equation of a pair of dependent linear equations is `-5x + 7y -2 = 0`. The second equation can be

Options

A. `10x + 14 y + 4 = 0`
B. `-10x - 14y +4 =0`
C. `-10 +14y +4 =0`
D. `10x - 14y + 4 = 0`

Explanation

To find a second equation that is dependent on the given equation \(-5x + 7y - 2 = 0\), we need to ensure that the two equations represent the same line. This means that the coefficients of \(x\) and \(y\) in the second equation must be proportional to those in the first equation.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ -5x + 7y - 2 = 0 \]
This can be rewritten in the standard form \(a_1x + b_1y + c_1 = 0\) where \(a_1 = -5\), \(b_1 = 7\), and \(c_1 = -2\).

2. **Identify the conditions for dependent equations**:
For two equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) to be dependent, the following must hold:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

3. **Choose a constant \(k\)**:
We can express the second equation as:
\[ a_2 = k \cdot a_1, \quad b_2 = k \cdot b_1, \quad c_2 = k \cdot c_1 \]
where \(k\) is any non-zero constant.

4. **Substituting the values**:
Let’s take \(k = -2\) (this is just an example; any non-zero value will work):
\[ a_2 = -2 \cdot (-5) = 10 \]
\[ b_2 = -2 \cdot 7 = -14 \]
\[ c_2 = -2 \cdot (-2) = 4 \]

5. **Form the second equation**:
Now we can write the second equation using these coefficients:
\[ 10x - 14y + 4 = 0 \]

6. **Conclusion**:
Thus, one possible second equation that is dependent on the first equation is:
\[ 10x - 14y + 4 = 0 \]