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:
`5x-4y+ 8=0`
`7x+ 6y-9=0`

Answer

To determine whether the lines represented by the equations \(5x - 4y + 8 = 0\) and \(7x + 6y - 9 = 0\) intersect at a point, are parallel, or are coincident, we will compare the ratios of the coefficients of the variables and the constant terms.

### Step-by-Step Solution:

1. **Rewrite the equations in standard form**:
- The first equation is already in standard form:
\[ 5x - 4y + 8 = 0 \quad \text{(Equation 1)} \]
- The second equation is also in standard form:
\[ 7x + 6y - 9 = 0 \quad \text{(Equation 2)} \]

2. **Identify coefficients**:
- For Equation 1, the coefficients are:
\[ a_1 = 5, \quad b_1 = -4, \quad c_1 = 8 \]
- For Equation 2, the coefficients are:
\[ a_2 = 7, \quad b_2 = 6, \quad c_2 = -9 \]

3. **Calculate the ratios**:
- Calculate \( \frac{a_1}{a_2} \):
\[ \frac{a_1}{a_2} = \frac{5}{7} \]
- Calculate \( \frac{b_1}{b_2} \):
\[ \frac{b_1}{b_2} = \frac{-4}{6} = \frac{-2}{3} \]
- Calculate \( \frac{c_1}{c_2} \):
\[ \frac{c_1}{c_2} = \frac{8}{-9} = -\frac{8}{9} \]

4. **Compare the ratios**:
- We have:
\[ \frac{a_1}{a_2} = \frac{5}{7}, \quad \frac{b_1}{b_2} = \frac{-2}{3}, \quad \frac{c_1}{c_2} = -\frac{8}{9} \]
- Now we check if the ratios are equal:
- \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) (since \( \frac{5}{7} \neq \frac{-2}{3} \))
- \( \frac{a_1}{a_2} \neq \frac{c_1}{c_2} \) (since \( \frac{5}{7} \neq -\frac{8}{9} \))
- \( \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) (since \( \frac{-2}{3} \neq -\frac{8}{9} \))

5. **Conclusion**:
- Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the lines represented by the equations are **intersecting lines**.