On comparing the ratios `(a_(1))/(a_(2)), (b_(1))/(b_(2)) and (c_(1))/(c_(2))`, find out whether the following pair of linear equations are consistent, or inconsistent.
`2x- 3y= 8, 4x- 6y=9`

Answer

To determine whether the given pair of linear equations is consistent or inconsistent, we will compare the ratios of their coefficients and constants.

The given equations are:
1. \( 2x - 3y = 8 \) (Equation 1)
2. \( 4x - 6y = 9 \) (Equation 2)

### Step 1: Rewrite the equations in the standard form
We can rewrite the equations in the form \( A_1x + B_1y + C_1 = 0 \) and \( A_2x + B_2y + C_2 = 0 \).

For Equation 1:
\[ 2x - 3y - 8 = 0 \implies A_1 = 2, B_1 = -3, C_1 = -8 \]

For Equation 2:
\[ 4x - 6y - 9 = 0 \implies A_2 = 4, B_2 = -6, C_2 = -9 \]

### Step 2: Calculate the ratios
Now, we will calculate the ratios \( \frac{A_1}{A_2} \), \( \frac{B_1}{B_2} \), and \( \frac{C_1}{C_2} \).

1. \( \frac{A_1}{A_2} = \frac{2}{4} = \frac{1}{2} \)
2. \( \frac{B_1}{B_2} = \frac{-3}{-6} = \frac{1}{2} \)
3. \( \frac{C_1}{C_2} = \frac{-8}{-9} = \frac{8}{9} \)

### Step 3: Compare the ratios
Now we compare the ratios:
- \( \frac{A_1}{A_2} = \frac{1}{2} \)
- \( \frac{B_1}{B_2} = \frac{1}{2} \)
- \( \frac{C_1}{C_2} = \frac{8}{9} \)

Since \( \frac{A_1}{A_2} = \frac{B_1}{B_2} \) but \( \frac{C_1}{C_2} \) is not equal to these ratios, we conclude that the equations are inconsistent.

### Conclusion
The given pair of linear equations is inconsistent because the ratios of the coefficients do not satisfy the condition for consistency.

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