On comparing the ratios `(a_(1))/(a_(2)), (b_(1))/(b_(2)) and (c_(1))/(c_(2))`, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
`6x- 3y+10=0`
`2x-y+ 9=0`

Answer

To determine whether the lines represented by the equations \(6x - 3y + 10 = 0\) and \(2x - y + 9 = 0\) intersect at a point, are parallel, or are coincident, we will compare the ratios of the coefficients of the variables and the constants.

### Step 1: Write the equations in standard form
The given equations are already in standard form:
1. \(6x - 3y + 10 = 0\)
2. \(2x - y + 9 = 0\)

### Step 2: Identify coefficients
From the first equation \(6x - 3y + 10 = 0\), we can identify:
- \(a_1 = 6\)
- \(b_1 = -3\)
- \(c_1 = 10\)

From the second equation \(2x - y + 9 = 0\), we can identify:
- \(a_2 = 2\)
- \(b_2 = -1\)
- \(c_2 = 9\)

### Step 3: Calculate the ratios
Now we will calculate the ratios:
1. \(\frac{a_1}{a_2} = \frac{6}{2} = 3\)
2. \(\frac{b_1}{b_2} = \frac{-3}{-1} = 3\)
3. \(\frac{c_1}{c_2} = \frac{10}{9}\)

### Step 4: Compare the ratios
Now we compare the calculated ratios:
- \(\frac{a_1}{a_2} = 3\)
- \(\frac{b_1}{b_2} = 3\)
- \(\frac{c_1}{c_2} = \frac{10}{9}\)

### Step 5: Analyze the results
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) but \(\frac{c_1}{c_2} \neq \frac{a_1}{a_2}\) (or \(\frac{b_1}{b_2}\)), we conclude that the lines are parallel.

### Final Conclusion
The lines represented by the equations \(6x - 3y + 10 = 0\) and \(2x - y + 9 = 0\) are parallel.

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