Do the equations `5x + 7 y =8 and 10 x + 14 y =4` represent a pair of coincident lines ? Justify your answer.

Answer

To determine whether the equations \(5x + 7y = 8\) and \(10x + 14y = 4\) represent a pair of coincident lines, we will check the condition for the coincidence of two lines.

### Step 1: Identify the coefficients and constants
For the equations:
1. \(5x + 7y = 8\)
- Coefficients: \(a_1 = 5\), \(b_1 = 7\), \(c_1 = 8\)
2. \(10x + 14y = 4\)
- Coefficients: \(a_2 = 10\), \(b_2 = 14\), \(c_2 = 4\)

### Step 2: Use the condition for coincident lines
The lines are coincident if:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

### Step 3: Calculate \(\frac{a_1}{a_2}\)
\[ \frac{a_1}{a_2} = \frac{5}{10} = \frac{1}{2} \]

### Step 4: Calculate \(\frac{b_1}{b_2}\)
\[ \frac{b_1}{b_2} = \frac{7}{14} = \frac{1}{2} \]

### Step 5: Calculate \(\frac{c_1}{c_2}\)
\[ \frac{c_1}{c_2} = \frac{8}{4} = 2 \]

### Step 6: Compare the ratios
Now we compare the values:
- \(\frac{a_1}{a_2} = \frac{1}{2}\)
- \(\frac{b_1}{b_2} = \frac{1}{2}\)
- \(\frac{c_1}{c_2} = 2\)

### Step 7: Conclusion
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) but \(\frac{c_1}{c_2}\) is not equal to these values, the condition for the lines to be coincident is not satisfied.

Thus, the equations \(5x + 7y = 8\) and \(10x + 14y = 4\) do not represent a pair of coincident lines.

### Final Answer
The given lines are not coincident.

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