Do the following equations represent a pair of coincidnet lines? Justify your answer.
`3x+ (1)/(7) y=3, 7x+ 3y= 7`

Answer

To determine if the given equations represent a pair of coincident lines, we will follow these steps:

### Given Equations:
1. \( 3x + \frac{1}{7}y = 3 \)
2. \( 7x + 3y = 7 \)

### Step 1: Rewrite the equations in standard form
The standard form of a linear equation is \( ax + by = c \).

- The first equation is already in standard form:
\[ 3x + \frac{1}{7}y = 3 \]

- The second equation can be rewritten as:
\[ 7x + 3y = 7 \]

### Step 2: Identify coefficients
From the equations, we identify:
- For the first equation \( a_1 = 3, b_1 = \frac{1}{7}, c_1 = 3 \)
- For the second equation \( a_2 = 7, b_2 = 3, c_2 = 7 \)

### Step 3: Calculate the ratios
To check if the lines are coincident, we need to verify the following condition:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

1. Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{3}{7} \]

2. Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{\frac{1}{7}}{3} = \frac{1}{21} \]

3. Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{3}{7} \]

### Step 4: Compare the ratios
Now we compare the calculated ratios:
- \( \frac{a_1}{a_2} = \frac{3}{7} \)
- \( \frac{b_1}{b_2} = \frac{1}{21} \)
- \( \frac{c_1}{c_2} = \frac{3}{7} \)

### Step 5: Conclusion
Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the ratios are not equal. Therefore, the two lines do not coincide.

### Final Answer:
No, the given equations do not represent a pair of coincident lines.

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