Do the equations `4x + 3y =6 and 12 x +9y =15` represent a pair of coincident lines ?

Answer

To determine whether the equations \(4x + 3y = 6\) and \(12x + 9y = 15\) represent a pair of coincident lines, we can use the condition for coincident lines. The equations of the lines can be expressed in the standard form \(A_1x + B_1y = C_1\) and \(A_2x + B_2y = C_2\).

### Step 1: Identify coefficients
From the given equations:
- For the first equation \(4x + 3y = 6\):
- \(A_1 = 4\)
- \(B_1 = 3\)
- \(C_1 = 6\)

- For the second equation \(12x + 9y = 15\):
- \(A_2 = 12\)
- \(B_2 = 9\)
- \(C_2 = 15\)

### Step 2: Set up the ratios
To check if the lines are coincident, we need to verify the following condition:
\[ \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} \]

### Step 3: Calculate the ratios
1. Calculate \(\frac{A_1}{A_2}\):
\[ \frac{A_1}{A_2} = \frac{4}{12} = \frac{1}{3} \]

2. Calculate \(\frac{B_1}{B_2}\):
\[ \frac{B_1}{B_2} = \frac{3}{9} = \frac{1}{3} \]

3. Calculate \(\frac{C_1}{C_2}\):
\[ \frac{C_1}{C_2} = \frac{6}{15} = \frac{2}{5} \]

### Step 4: Compare the ratios
Now we compare the calculated ratios:
- \(\frac{A_1}{A_2} = \frac{1}{3}\)
- \(\frac{B_1}{B_2} = \frac{1}{3}\)
- \(\frac{C_1}{C_2} = \frac{2}{5}\)

Since \(\frac{A_1}{A_2} = \frac{B_1}{B_2}\) but \(\frac{C_1}{C_2}\) is not equal to these, we conclude that:

### Conclusion
The equations \(4x + 3y = 6\) and \(12x + 9y = 15\) do not represent a pair of coincident lines.

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