For which value of the parameter g do the linear equations 6x + 12y + 9 = 0 and 2x + g y + 3 = 0 represent the same straight line, that is, have infinitely many common solutions?

Introduction / Context:
This question checks understanding of when two linear equations in x and y represent the same straight line. If they represent the same line, then every point on one line lies on the other, meaning the equations are proportional. This is a standard topic in coordinate geometry and linear equations in two variables.

Given Data / Assumptions:

• The first line is given by 6x + 12y + 9 = 0.
• The second line is 2x + g y + 3 = 0, where g is a real parameter.
• We must find g such that these two equations describe exactly the same geometric line.
• All coefficients are real numbers.

Concept / Approach:
Two linear equations a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 represent the same straight line if and only if there exists a non zero constant k such that a1 = k a2, b1 = k b2, and c1 = k c2. In other words, the ratios a1/a2, b1/b2, and c1/c2 are all equal. We apply this idea to the given pair of equations and solve for g so that the three ratios match.

Step-by-Step Solution:
Write the equations in coefficient form: For the first line, a1 = 6, b1 = 12, c1 = 9.For the second line, a2 = 2, b2 = g, c2 = 3.If the lines are the same, then 6/2 = 12/g = 9/3.Compute 6/2 = 3 and 9/3 = 3, so the common ratio must be 3.Set 12/g equal to 3: 12/g = 3.Solve for g: multiply both sides by g to get 12 = 3g, so g = 12/3 = 4.

Verification / Alternative check:
With g = 4, the second equation becomes 2x + 4y + 3 = 0. Multiply this entire equation by 3 to see if it matches the first: 3 * (2x + 4y + 3) = 6x + 12y + 9 = 0. This is exactly the first equation. Therefore, for g = 4, each equation is a constant multiple of the other, and they represent the same line. Any other value of g would make the ratios of corresponding coefficients unequal and hence describe a different line or a line that intersects at a single point.

Why Other Options Are Wrong:

• For g = 3, 6/2 = 3 but 12/3 = 4, so the ratios are not equal.
• For g = 6, 12/6 = 2, which does not match 6/2 = 3.
• For g = 9, 12/9 = 4/3, again different from 3.
• For g = 0, the second equation becomes 2x + 3 = 0, which is a vertical shift and not a multiple of the first equation.

Common Pitfalls:

• Equating only two of the ratios and forgetting to check the third constant term, which must also match the same ratio.
• Mixing up the order of coefficients when forming ratios a1/a2, b1/b2, and c1/c2.
• Assuming that having equal slopes alone is enough; equal slopes give parallel lines, but equal ratios including constants give exactly the same line.

Final Answer:
4