For the straight line represented by the equation 3x - 6y = 4, what is the slope of a line that is perpendicular to this line?

Introduction / Context:
In coordinate geometry, the slope of a line describes its steepness and direction. Many aptitude questions test your understanding of how slopes of parallel and perpendicular lines are related. In this problem, you are given the equation of one line and asked to find the slope of a line that is perpendicular to it. This checks both algebraic manipulation of a linear equation and the geometric concept of perpendicularity between lines.

Given Data / Assumptions:

• The given line has equation 3x - 6y = 4.
• We assume this represents a non vertical straight line in the xy plane.
• We need the slope of any line that is perpendicular to this line.
• Standard slope intercept form is y = mx + c, where m is the slope.

Concept / Approach:
The main ideas are that any linear equation can be rewritten in slope intercept form, and that the product of the slopes of two perpendicular non vertical lines is -1. If a line has slope m, then a line perpendicular to it has slope equal to the negative reciprocal, which is -1/m. So we will first rewrite the given equation to find its slope, and then use the negative reciprocal relationship to determine the slope of the perpendicular line.

Step-by-Step Solution:
Step 1: Start from the given equation 3x - 6y = 4. Step 2: Rearrange to solve for y in terms of x: -6y = 4 - 3x. Step 3: Divide both sides by -6 to isolate y: y = (3/6)x - 4/6. Step 4: Simplify the fractions: y = (1/2)x - 2/3. Therefore the slope of the given line is 1/2. Step 5: For a perpendicular line, slope m2 must satisfy m1 * m2 = -1. With m1 = 1/2, we get (1/2) * m2 = -1, so m2 = -2.

Verification / Alternative check:
We can quickly verify by checking the product of slopes. The original line has slope 1/2. The candidate perpendicular slope is -2. Their product is (1/2) * (-2) = -1, which confirms the lines are perpendicular in Euclidean geometry. This simple numeric check is enough to validate that the answer is consistent with the rule for perpendicular lines in the coordinate plane.

Why Other Options Are Wrong:

• Option b (2) is the negative reciprocal of -1/2, not of 1/2, so it would be perpendicular to a line with slope -1/2, not 1/2.
• Option c (2/3) is neither equal to 1/2 nor its negative reciprocal, so it describes a line with a completely different orientation.
• Option d (-2/3) is negative but its product with 1/2 is -1/3, not -1, so it is not perpendicular.
• Option e (1/2) is the slope of the original line itself, which would give a parallel line rather than a perpendicular one.

Common Pitfalls:
Students sometimes forget to convert the given line into slope intercept form correctly and may make sign or fraction mistakes in the algebra. Another frequent error is thinking that the perpendicular slope is simply the negative of the original slope instead of the negative reciprocal. It is also common to misinterpret parallel and perpendicular relationships, especially when in a hurry during an exam. Careful rearrangement and use of the product equals negative one rule avoids these issues.

Final Answer:
The slope of the line perpendicular to 3x - 6y = 4 is -2.