The straight line ax − 4y = −6 has slope −3/2. What is the value of the coefficient a?

Introduction / Context:
This question tests your understanding of how to read the slope of a line from its equation in general form. Converting a linear equation to slope intercept form y = mx + c makes it easy to identify the slope m and relate it to given information. This is a core skill in coordinate geometry.

Given Data / Assumptions:

• The equation of the line is ax − 4y = −6.
• The slope of this line is given as −3 / 2.
• We must find the value of a that makes the slope equal to −3 / 2.
• We assume the usual slope intercept interpretation: slope is the coefficient of x when the equation is written as y = mx + c.

Concept / Approach:
To find the slope, we rewrite the line in the form y = mx + c. The coefficient m of x is the slope. Starting from ax − 4y = −6, solve for y in terms of x. The resulting expression will involve a, and equating its coefficient of x to −3 / 2 will allow us to solve for a. This is straightforward algebra once the equation is rearranged correctly.

Step-by-Step Solution:
Step 1: Start with the given equation ax − 4y = −6. Step 2: Isolate the y term: −4y = −ax − 6. Step 3: Divide both sides by −4 to solve for y: y = (−ax − 6) / (−4) = (ax + 6) / 4. Step 4: Rewrite in the form y = mx + c: y = (a / 4)x + 6 / 4. Step 5: The slope m of the line is therefore a / 4. Step 6: We are told the slope is −3 / 2, so set a / 4 = −3 / 2. Step 7: Solve for a: multiply both sides by 4 to get a = 4 * (−3 / 2) = −12 / 2 = −6.

Verification / Alternative check:
Substitute a = −6 back into the equation to check the slope. The line becomes −6x − 4y = −6. Rearranging, −4y = 6x − 6, so y = (−6 / 4)x + (6 / 4) = (−3 / 2)x + 3 / 2. The coefficient of x is indeed −3 / 2, confirming that the slope condition is satisfied and that a = −6 is correct.

Why Other Options Are Wrong:
If a = 6, 3, −3 or 0, the slope becomes a / 4 = 3 / 2, 3 / 4, −3 / 4 or 0 respectively, none of which equal −3 / 2. Each alternative value for a changes the tilt of the line, so these options do not match the given slope requirement.

Common Pitfalls:
Errors often come from sign mistakes when moving terms across the equal sign or dividing by negative numbers. Another common issue is misreading the general form and thinking the slope is directly a or −a / 4 without performing the proper algebra. Always explicitly rearrange to y = mx + c to avoid confusion and double check the sign of each term.

Final Answer:
The value of a is −6.