In coordinate geometry, find the slope of any straight line that is parallel to the line 3x − 6y = 4.

Introduction / Context:
This question checks your understanding of slopes of straight lines and the condition for two lines to be parallel. In coordinate geometry, lines that are parallel share the same slope, even though their intercepts on the axes may be different. Being able to read the slope from a linear equation and transfer that knowledge to other lines is a core skill in analytic geometry and is very useful in many aptitude exams.

Given Data / Assumptions:
- We are given the line 3x − 6y = 4.
- We are asked for the slope of any line that is parallel to this line.
- The coordinate system is standard, with x and y axes perpendicular to each other.

Concept / Approach:
A line written in the form y = mx + c has slope m. For a general equation ax + by + c = 0, we can rearrange to y = (−a/b)x − c/b to identify the slope. Two non vertical lines are parallel if and only if their slopes are equal. Therefore we simply need to find the slope of the given line and use that as the answer for all parallel lines.

Step-by-Step Solution:
Step 1: Start with the given equation 3x − 6y = 4. Step 2: Rearrange it to express y in terms of x. Subtract 3x from both sides to obtain −6y = −3x + 4. Step 3: Divide every term by −6 to isolate y, giving y = (−3x + 4) / (−6). Step 4: Simplify the fractions: y = (3/6)x − 4/6 = (1/2)x − 2/3. Step 5: The equation is now in the form y = mx + c with m = 1/2, so the slope of the given line is 1/2. Step 6: Any line parallel to this one must have exactly the same slope, namely 1/2.
Verification / Alternative check:
You can check the result quickly by picking any two points on the line. For example, when x = 2, y = (1/2)*2 − 2/3 = 1 − 2/3 = 1/3. When x = 4, y = 2 − 2/3 = 4/3. The slope between these two points is (4/3 − 1/3) / (4 − 2) = (3/3) / 2 = 1/2, which matches our algebraic derivation.

Why Other Options Are Wrong:
Option a, −1/2, corresponds to the slope of a line that would be symmetric in gradient but not parallel.
Option c, 2, is the negative reciprocal of −1/2 and would give a perpendicular line, not a parallel one.
Option d, −2, and option e, 0, represent different slopes altogether and do not satisfy the condition of parallelism with the original line.

Common Pitfalls:
Students sometimes forget to divide all terms correctly when rearranging the equation, which can lead to sign mistakes in the slope. Another frequent error is to misinterpret the constant term as affecting the slope, while only the coefficient of x determines the slope in the form y = mx + c. Careful algebra and keeping track of negative signs prevents these errors.

Final Answer:
The slope of any line parallel to 3x − 6y = 4 is 1/2.