Is the pair of equations `3x - 5y =6 and 4x -6y =7` consistent ? Justify your answer.

Answer

To determine if the pair of equations \(3x - 5y = 6\) and \(4x - 6y = 7\) is consistent, we can use the concept of the ratio of coefficients.

### Step-by-Step Solution:

1. **Write the equations in standard form:**
The given equations are:
\[ 3x - 5y - 6 = 0 \quad \text{(Equation 1)} \]
\[ 4x - 6y - 7 = 0 \quad \text{(Equation 2)} \]

2. **Identify coefficients:**
From the equations, we can identify the coefficients:
- For Equation 1: \(A_1 = 3\), \(B_1 = -5\), \(C_1 = -6\)
- For Equation 2: \(A_2 = 4\), \(B_2 = -6\), \(C_2 = -7\)

3. **Calculate the ratios of the coefficients:**
We need to check the ratios \( \frac{A_1}{A_2} \) and \( \frac{B_1}{B_2} \):
\[ \frac{A_1}{A_2} = \frac{3}{4} \]
\[ \frac{B_1}{B_2} = \frac{-5}{-6} = \frac{5}{6} \]

4. **Check if the ratios are equal:**
We compare the two ratios:
\[ \frac{3}{4} \neq \frac{5}{6} \]
Since the ratios are not equal, we can conclude that the lines represented by the equations are not parallel.

5. **Conclusion:**
Since the lines are not parallel, they must intersect at a unique point. Therefore, the pair of equations is consistent.

### Final Answer:
The pair of equations \(3x - 5y = 6\) and \(4x - 6y = 7\) is consistent because the ratios of the coefficients \( \frac{A_1}{A_2} \) and \( \frac{B_1}{B_2} \) are not equal, indicating that the lines intersect at a unique point.