For the straight line 3x + y = -6, at which point does it intersect the x-axis?

Introduction / Context:
This question is about finding the x intercept of a straight line given in standard linear form. The x intercept is the point where the line crosses the x axis, and at this point the y coordinate is zero. Understanding x and y intercepts is a basic but very important concept in coordinate geometry, forming the foundation for graphing lines, solving linear equations, and interpreting linear relationships in many applied contexts such as economics, physics, and data analysis.

Given Data / Assumptions:
• The equation of the line is 3x + y = -6.
• The x axis consists of all points where y = 0.
• The x intercept is the point on the line whose y coordinate is zero.
• We want the coordinates in the form (x, 0).

Concept / Approach:
To find the x intercept of a line, we substitute y = 0 into the equation and then solve for x. This works because all points on the x axis have y equal to zero. Once x is determined, we can form the ordered pair (x, 0), which gives the exact location where the line crosses the x axis. This method is straightforward for any linear equation in two variables.

Step-by-Step Solution:
Step 1: Start from the given line equation: 3x + y = -6. Step 2: At the x axis, the y coordinate equals zero, so set y = 0. Step 3: Substitute y = 0 into the equation to obtain 3x + 0 = -6. Step 4: Simplify this to 3x = -6. Step 5: Solve for x by dividing both sides by 3, giving x = -6 / 3 = -2. Step 6: Therefore, the x intercept is the point with coordinates (-2, 0).

Verification / Alternative check:
To verify, we can substitute x = -2 and y = 0 back into the original equation. The left hand side becomes 3 * (-2) + 0 = -6. The right hand side is also -6, so the equation is satisfied. This confirms that (-2, 0) lies on the line. Further, this point clearly has y = 0, which means it lies on the x axis as well. Since it lies on both the line and the x axis, it must be the x intercept. None of the other listed points satisfy both conditions simultaneously.

Why Other Options Are Wrong:
(2, 0) gives 3 * 2 + 0 = 6, which does not equal -6, so it is not on the line.
(0, -2) has y = -2, so it is not on the x axis, and substituting gives 3 * 0 + (-2) = -2, not -6.
(0, -6) is on the y axis, not the x axis, and represents the y intercept instead.
(-3, 0) produces 3 * (-3) + 0 = -9, which does not match the required right hand side of -6. Thus all these options are invalid.

Common Pitfalls:
One common error is to confuse the x intercept with the y intercept and substitute x = 0 instead of y = 0. Another mistake is to miscalculate the division step, for example dividing -6 by 3 incorrectly. Students may also choose points that satisfy the equation but are not on the x axis, or vice versa. To avoid these issues, always remember that x intercept means y = 0, and double check the substitution by plugging the candidate point back into the original equation.

Final Answer:
The line 3x + y = -6 intersects the x axis at the point (-2, 0).