Assertion (A) :Pair of linear equations `9x+3y+12=0` and `18x+6y+24=0` have infinitely many solutions
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 and reason provided are true, we will analyze the pair of linear equations step by step.

### Step 1: Identify the equations
The given equations are:
1. \( 9x + 3y + 12 = 0 \)
2. \( 18x + 6y + 24 = 0 \)

### Step 2: Rewrite the equations in standard form
We can rewrite both equations in the standard form \( ax + by + c = 0 \):
1. \( 9x + 3y + 12 = 0 \) can be rewritten as \( 9x + 3y = -12 \).
2. \( 18x + 6y + 24 = 0 \) can be rewritten as \( 18x + 6y = -24 \).

### Step 3: Identify coefficients
From the equations, we can identify the coefficients:
- For the first equation: \( a_1 = 9, b_1 = 3, c_1 = 12 \)
- For the second equation: \( a_2 = 18, b_2 = 6, c_2 = 24 \)

### Step 4: Check the ratios
To check if the equations have infinitely many solutions, we need to verify if the following ratios are equal:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

Calculating the ratios:
- \( \frac{a_1}{a_2} = \frac{9}{18} = \frac{1}{2} \)
- \( \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \)
- \( \frac{c_1}{c_2} = \frac{12}{24} = \frac{1}{2} \)

### Step 5: Conclusion about the assertion
Since all three ratios are equal, we conclude that the equations \( 9x + 3y + 12 = 0 \) and \( 18x + 6y + 24 = 0 \) have infinitely many solutions. Therefore, the assertion (A) is true.

### Step 6: Conclusion about the reason
The reason (R) states that a pair of linear equations has infinitely many solutions if the ratios \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) are equal. Since we have verified this condition holds true, the reason (R) is also true.

### Final Conclusion
Both the assertion (A) and the reason (R) are true, and the reason is the correct explanation of the assertion.