In coordinate geometry, what is the equation of a straight line that has slope −1/3 and a y-intercept equal to 6?

Introduction / Context: This question tests the relationship between the slope intercept form of a line and its general form. Being able to move between y = mx + c and standard forms such as ax + by + c = 0 is a foundational skill in coordinate geometry and appears frequently in aptitude and competitive examinations.

Given Data / Assumptions:

• Slope of the required line, m = −1/3.
• Y intercept, which is the point where the line crosses the y axis, is 6.
• We seek a correct equation among the options that represents this line.

Concept / Approach: The slope intercept form of a straight line is y = mx + c, where m is the slope and c is the y intercept. Once we write the line in this form, we can rearrange it into the general form ax + by + d = 0 or equivalently ax + by = k for comparison with the answer choices. Algebraic manipulation like multiplying throughout by a constant does not change the set of points that lie on the line.

Step-by-Step Solution: Write the equation in slope intercept form: y = (−1/3)x + 6. Multiply both sides by 3 to eliminate the denominator: 3y = −x + 18. Rearrange the terms to put x on the left side: x + 3y = 18. This is now in the same structural form as the multiple choice options. Compare with the given options to identify which equation matches x + 3y = 18 exactly.

Verification / Alternative check: We can quickly check by finding the intercepts. For x + 3y = 18, setting x = 0 gives 3y = 18, so y = 6, which matches the required y intercept. For the slope, rewrite as 3y = −x + 18, giving y = (−1/3)x + 6, where the slope is indeed −1/3. No other option yields both slope −1/3 and y intercept 6 when rearranged into y = mx + c form.

Why Other Options Are Wrong: x − 3y = 6 gives y = (1/3)x − 2, which has slope 1/3 and intercept −2, both incorrect. x + 3y = −18 gives y intercept −6. x − 3y = −6 gives y = (1/3)x + 2, again wrong slope and intercept. The option 3x + y = 6 gives y = −3x + 6, slope −3 instead of −1/3. Thus none of these match the stated slope and intercept.

Common Pitfalls: A frequent error is to invert the slope or to confuse the roles of x and y intercepts. Another issue is forgetting that multiplying the entire equation by a nonzero constant does not change the line, but changing signs or moving terms incorrectly can alter the relationship between slope and intercept. Always return to y = mx + c to check slope and intercept clearly.

Final Answer: The required line has equation x + 3y = 18.