To solve this problem, we have two linear equations given:

1. $3x + 2y + 4 = 0$
2. $6x + 4y + c = 0$

The question is asking for the value of $c$ such that these two equations represent the same line.

Step-by-Step Solution

For two lines to be the same, their equations must be proportional. This means the coefficients of $x$, $y$, and the constant term should be in the same ratio.

Let's compare the two equations term by term:

1. $\frac{6}{3} = 2$
2. $\frac{4}{2} = 2$
3. $\frac{c}{4}$

Since the two lines are identical, all three ratios must be equal. Thus,

$\frac{c}{4} = 2$

Solving for $c$:

$c = 2 \times 4 = 8$

Answer

The value of $c$ is $8$.

Would you like further details on any step, or have additional questions?

Here are five related questions for further practice:

1. What if the lines were parallel but not identical? How would the value of $c$ change?
2. How would you determine if two lines are perpendicular given their equations?
3. What is the general form of a line equation, and how can it be converted to slope-intercept form?
4. Can two lines with different slopes ever represent the same line?
5. How can we find the point of intersection for two lines given in standard form?

Tip: When comparing linear equations for similarity, always check if the coefficients are proportional to confirm that they represent the same line.