Is the pair of equations `x+2y-3=0 and 6y+3x-9=0` consistent? Justify your answer.

Answer

To determine if the pair of equations \( x + 2y - 3 = 0 \) and \( 6y + 3x - 9 = 0 \) is consistent, we can follow these steps:

### Step 1: Rewrite the equations in standard form
The first equation is already in standard form:
1. \( x + 2y - 3 = 0 \)

For the second equation, we can rearrange it:
2. \( 6y + 3x - 9 = 0 \) can be rewritten as \( 3x + 6y - 9 = 0 \).

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:
- For the first equation \( a_1 = 1, b_1 = 2, c_1 = -3 \)
- For the second equation \( a_2 = 3, b_2 = 6, c_2 = -9 \)

### Step 3: Calculate the ratios
Now we will calculate the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \):
- \( \frac{a_1}{a_2} = \frac{1}{3} \)
- \( \frac{b_1}{b_2} = \frac{2}{6} = \frac{1}{3} \)
- \( \frac{c_1}{c_2} = \frac{-3}{-9} = \frac{1}{3} \)

### Step 4: Analyze the ratios
Since we have:
- \( \frac{a_1}{a_2} = \frac{1}{3} \)
- \( \frac{b_1}{b_2} = \frac{1}{3} \)
- \( \frac{c_1}{c_2} = \frac{1}{3} \)

All three ratios are equal, which indicates that the two equations represent the same line.

### Step 5: Conclusion
Since the two equations represent the same line, they have infinitely many solutions. Therefore, the pair of equations is consistent.

### Summary
The pair of equations \( x + 2y - 3 = 0 \) and \( 6y + 3x - 9 = 0 \) is consistent because they represent the same line, leading to infinitely many solutions.

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