Two statements are given below - one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).
Assertion (A): `9 x + 12 y - 7 = 0` and` 6 x + 8 y - 14 = 0` form a consistent pair of linear equations.
Assertion (A): `9 x + 12 y - 7 = 0` and `6 x + 8 y - 14 = 0` form a consistent pair of linear equations. `(p)/(f)=(q)/(g) ne (r)/(h)`

Options

A. Both (A) and (R) are true and (R) is the correct explanation of the (A).
B. Both (A) and (R) are true but (R) is not the correct explanation of the (A).
C. (A) is true but (R) is false
D. (A) is false but (R) is true.

Answer

To determine whether the two given linear equations form a consistent pair, we need to analyze the equations and their relationships. The equations given are:

1. \(9x + 12y - 7 = 0\)
2. \(6x + 8y - 14 = 0\)

### Step 1: Rewrite the equations in standard form
We can rewrite these equations in the form \(Ax + By = C\):

1. \(9x + 12y = 7\) (Equation 1)
2. \(6x + 8y = 14\) (Equation 2)

### Step 2: Identify coefficients
From the equations, we can identify the coefficients:

- For Equation 1: \(a_1 = 9\), \(b_1 = 12\), \(c_1 = 7\)
- For Equation 2: \(a_2 = 6\), \(b_2 = 8\), \(c_2 = 14\)

### Step 3: Check the ratios
To determine if the equations are consistent or inconsistent, we need to check the ratios:

1. Calculate \( \frac{a_1}{a_2} = \frac{9}{6} = \frac{3}{2} \)
2. Calculate \( \frac{b_1}{b_2} = \frac{12}{8} = \frac{3}{2} \)
3. Calculate \( \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2} \)

### Step 4: Analyze the results
Now we compare the ratios:

- \( \frac{a_1}{a_2} = \frac{3}{2} \)
- \( \frac{b_1}{b_2} = \frac{3}{2} \)
- \( \frac{c_1}{c_2} = \frac{1}{2} \)

We observe that:

- \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) (both are equal to \( \frac{3}{2} \))
- \( \frac{c_1}{c_2} \) is not equal to \( \frac{a_1}{a_2} \) or \( \frac{b_1}{b_2} \) (it is \( \frac{1}{2} \))

### Step 5: Conclusion
According to the conditions for consistency:

- The equations are consistent if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) and \( \frac{c_1}{c_2} \) is also equal to these ratios.
- The equations are inconsistent if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \frac{c_1}{c_2} \) is not equal.

Since we found that \( \frac{c_1}{c_2} \) is not equal to \( \frac{a_1}{a_2} \) or \( \frac{b_1}{b_2} \), the two equations do not form a consistent pair.

### Final Answer
Both statements (Assertion and Reason) are incorrect. The correct option is that both statements are false.

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