In coordinate geometry, the straight line a * x + 3y = 6 is given to have slope −2/3. Using the relationship between the slope and the coefficients in the standard form of a line, find the value of the constant a.

Introduction / Context:
This question checks understanding of how to read the slope of a line from its equation. The line is given in the general form a x + 3y = 6, and you are told the numerical value of its slope. By converting the equation into slope intercept form y = m x + c, you can compare coefficients and solve for the unknown constant a. This is a basic but fundamental skill in analytic geometry.

Given Data / Assumptions:

• The equation of the line is a x + 3y = 6.
• The slope of this line is given as −2/3.
• a is an unknown constant that we must determine.
• The coordinate system is the standard Cartesian plane with x and y axes.

Concept / Approach:
A line in the form A x + B y + C = 0 can be rewritten in slope intercept form y = m x + c, where m is the slope. Alternatively, for a line written as A x + B y + C = 0, the slope m is equal to −A / B, provided B is not zero. Here, a x + 3y = 6 can be seen as A x + B y + C = 0 with A = a, B = 3, and C = −6. Using either approach, we can express the slope in terms of a and set it equal to the given slope −2/3.

Step-by-Step Solution:
1) Start with the equation a x + 3y = 6. 2) Rewrite in slope intercept form by solving for y in terms of x. 3) Move the a x term to the right side: 3y = −a x + 6. 4) Divide both sides by 3 to isolate y: y = (−a / 3) x + 2. 5) Compare with the standard slope intercept form y = m x + c, where m is the slope. 6) The slope of this line is therefore m = −a / 3. 7) We are given that the slope is −2/3, so set −a / 3 = −2 / 3. 8) Multiply both sides by −3 to solve for a: a = 2.

Verification / Alternative check:
Substitute a = 2 back into the equation of the line to check the slope. The equation becomes 2x + 3y = 6. Solving for y gives 3y = −2x + 6, so y = (−2/3)x + 2. The coefficient of x is −2/3, which matches the given slope exactly. This confirms that a = 2 is the correct value. Using the alternative formula slope m = −A / B with A = 2 and B = 3 also gives m = −2/3.

Why Other Options Are Wrong:
Option b (−2) would give a line −2x + 3y = 6, whose slope is m = 2/3, not −2/3. Option c (3) leads to a slope of −3/3 = −1. Option d (−3) leads to a slope of 3/3 = 1. Option e (0) gives the line 3y = 6, or y = 2, which has zero slope and is horizontal. None of these match the required slope of −2/3. Only option a produces the correct slope.

Common Pitfalls:
A common mistake is to mix up the signs when using the slope formula m = −A / B, or to incorrectly divide when converting to slope intercept form. Another error is mis reading the given slope and solving for 3/a instead of −a/3. Writing the intermediate step y = (−a / 3)x + 2 clearly and comparing it carefully to y = m x + c helps ensure the correct identification of the slope and avoids sign errors.

Final Answer:
The constant a must equal 2 so that the line a x + 3y = 6 has slope −2/3.