For which value of g will the two linear equations 6x + 12y + 9 = 0 and 2x + g y + 3 = 0 represent the same straight line with infinitely many common solutions? Choose the correct value of g.

Introduction / Context:
This question tests your understanding of when two linear equations in two variables represent the same straight line. For two equations to describe the same line, all their corresponding coefficients must be in the same constant ratio. Here you are asked to find the value of g that makes the second equation a scalar multiple of the first.

Given Data / Assumptions:

• First equation: 6x + 12y + 9 = 0.
• Second equation: 2x + g y + 3 = 0.
• Both are linear equations in x and y.
• We want them to represent the same line, not just intersecting or parallel distinct lines.

Concept / Approach:
For two lines a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 to represent the same line, there must exist a constant k such that a1/a2 = b1/b2 = c1/c2 = k. We use this condition to determine g by comparing coefficients of x, y, and the constant term in both equations.

Step-by-Step Solution:
Compare coefficients of x: 6 in the first equation and 2 in the second. The ratio is 6/2 = 3. Compare constant terms: 9 in the first equation and 3 in the second. The ratio is 9/3 = 3, consistent with the previous ratio. To represent the same line, the ratio of y coefficients must also be 3: 12 / g = 3. Solve 12 / g = 3 to get g = 12 / 3 = 4.
Verification / Alternative check:
You can rewrite the second equation with g = 4 as 2x + 4y + 3 = 0. Multiplying this entire equation by 3 gives 6x + 12y + 9 = 0, which is exactly the first equation. This confirms that with g = 4, the two equations are scalar multiples and therefore represent the same line.

Why Other Options Are Wrong:
Options a, b, d, and e give values of g that do not satisfy the proportionality condition. For example, if g = 6, then 12/6 = 2, which does not match the x and constant ratios of 3. Similarly, g = 3 or 9 or 12 also break the equality of ratios, so the corresponding lines would not coincide exactly.

Common Pitfalls:
Learners sometimes equate individual coefficients directly instead of using ratios, which only works if both equations are already in the same normalised form. Another common mistake is to match only the x and y coefficients and ignore the constants, which can lead to parallel but distinct lines rather than the same line.

Final Answer:
The two linear equations represent the same straight line when g = 4.