Assertion (A) : The lines `2x-5y=7` and `6x-15y=8` are parallel lines.
Reason (R ) : Pair of linear equations `a_(1)x+b_(1)y+c_(1)=0` and `a_(2)x+b_(2)y+c_(2)=0` have infinitely many solutions if `(a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`

Options

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

Answer

To determine whether the assertion (A) and reason (R) are true or false, we will analyze the equations given and the conditions for parallel lines.

### Step 1: Identify the equations
The equations given are:
1. \(2x - 5y = 7\) (Equation 1)
2. \(6x - 15y = 8\) (Equation 2)

### Step 2: Convert the equations to standard form
We can rewrite both equations in the standard form \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\).

1. For Equation 1:
\[ 2x - 5y - 7 = 0 \quad \Rightarrow \quad a_1 = 2, \, b_1 = -5, \, c_1 = -7 \]

2. For Equation 2:
\[ 6x - 15y - 8 = 0 \quad \Rightarrow \quad a_2 = 6, \, b_2 = -15, \, c_2 = -8 \]

### Step 3: Check the condition for parallel lines
For two lines to be parallel, the ratios of their coefficients must be equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \quad \text{and} \quad \frac{c_1}{c_2} \text{ should not be equal to the above ratios.} \]

Calculating the ratios:
\[ \frac{a_1}{a_2} = \frac{2}{6} = \frac{1}{3} \]
\[ \frac{b_1}{b_2} = \frac{-5}{-15} = \frac{1}{3} \]
\[ \frac{c_1}{c_2} = \frac{-7}{-8} = \frac{7}{8} \]

### Step 4: Analyze the results
Since:
\[ \frac{a_1}{a_2} = \frac{1}{3}, \quad \frac{b_1}{b_2} = \frac{1}{3}, \quad \text{and} \quad \frac{c_1}{c_2} = \frac{7}{8} \]
We find that:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \quad \text{but} \quad \frac{c_1}{c_2} \text{ is not equal to } \frac{a_1}{a_2} \text{ or } \frac{b_1}{b_2}. \]
Thus, the lines are parallel.

### Step 5: Evaluate the assertion (A) and reason (R)
- **Assertion (A)**: True (the lines are parallel).
- **Reason (R)**: The condition for infinitely many solutions is not satisfied since \(\frac{c_1}{c_2} \neq \frac{a_1}{a_2}\) and \(\frac{b_1}{b_2}\). Therefore, the reason is false.

### Conclusion
- Assertion (A) is true.
- Reason (R) is false.