Do the equations 3x + 6y = 2 and 6x + 12y = 4 represent a pair of consistent lines ?

Answer

To determine if the equations \(3x + 6y = 2\) and \(6x + 12y = 4\) represent a pair of consistent lines, we will analyze the coefficients of the equations and check the conditions for consistency.

### Step 1: Write the equations in standard form
The given equations are:
1. \(3x + 6y = 2\)
2. \(6x + 12y = 4\)

### Step 2: Identify coefficients
From the first equation \(3x + 6y - 2 = 0\), we can identify:
- \(a_1 = 3\)
- \(b_1 = 6\)
- \(c_1 = -2\)

From the second equation \(6x + 12y - 4 = 0\), we can identify:
- \(a_2 = 6\)
- \(b_2 = 12\)
- \(c_2 = -4\)

### Step 3: Calculate the ratios
Now we will calculate the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).

1. Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{3}{6} = \frac{1}{2} \]

2. Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{6}{12} = \frac{1}{2} \]

3. Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{-2}{-4} = \frac{1}{2} \]

### Step 4: Analyze the ratios
We have:
- \( \frac{a_1}{a_2} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{1}{2} \)
- \( \frac{c_1}{c_2} = \frac{1}{2} \)

Since all three ratios are equal, we can conclude that the two equations represent the same line.

### Step 5: Conclusion
The equations \(3x + 6y = 2\) and \(6x + 12y = 4\) are consistent and represent an infinite number of solutions because they are equivalent.

### Final Answer
Yes, the equations \(3x + 6y = 2\) and \(6x + 12y = 4\) represent a pair of consistent lines with infinite solutions.

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