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 can analyze the coefficients of the equations.

### Step 1: Identify the coefficients
The given equations can be written in the standard form \(Ax + By + C = 0\).
- For the first equation \(6x - 3y + 10 = 0\):
- \(A_1 = 6\)
- \(B_1 = -3\)
- \(C_1 = 10\)

- For the second equation \(2x - y + 9 = 0\):
- \(A_2 = 2\)
- \(B_2 = -1\)
- \(C_2 = 9\)

### Step 2: Calculate the ratios of the coefficients
Now we will calculate the ratios of the coefficients:
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 3: Analyze the ratios
We have:
- \(\frac{A_1}{A_2} = 3\)
- \(\frac{B_1}{B_2} = 3\)
- \(\frac{C_1}{C_2} = \frac{10}{9}\)

### Step 4: Determine the relationship between the lines
According to the conditions for the relationship between two lines:
- If \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\), the lines are parallel.
- If all three ratios are equal, the lines are coincident.
- If \(\frac{A_1}{A_2} \neq \frac{B_1}{B_2}\), the lines intersect at a point.

In our case:
- \(\frac{A_1}{A_2} = \frac{B_1}{B_2} = 3\) and \(\frac{C_1}{C_2} \neq 3\) (since \(\frac{10}{9} \neq 3\)).

### Conclusion
Since \(\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}\), the lines are parallel.