Find the point where the line 2x + 5y = -6 cuts the x-axis by setting y = 0, and then choose the correct x-intercept coordinate from the options.

Introduction / Context:
This coordinate geometry problem asks you to find the x-intercept of a line given in standard form. The x-intercept is the point where the line crosses the x-axis. This concept appears frequently in graph based questions and helps you draw quick sketches of lines or interpret their behaviour by looking at how they intersect the axes.

Given Data / Assumptions:

• The equation of the line is 2x + 5y = -6.
• The x-axis is defined by y = 0.
• The x-intercept has coordinates (x, 0).
• We need to solve for x when y is set to 0.

Concept / Approach:
To find the x-intercept of a line, we set y = 0 in the equation, because any point on the x-axis has a y coordinate of zero. Substituting y = 0 into the given equation reduces it to a simple linear equation in x. Solving this gives the x coordinate of the intercept. The resulting point (x, 0) is the required x-intercept.

Step-by-Step Solution:
Start from the line equation: 2x + 5y = -6. For the x-axis, set y = 0. Substitute y = 0: 2x + 5(0) = -6, which simplifies to 2x = -6. Solve for x: x = -6 / 2 = -3. Thus, the line intersects the x-axis at the point (-3, 0).
Verification / Alternative check:
You can also rewrite the equation in intercept form if desired. Solve for y to get 5y = -6 - 2x, so y = (-6 - 2x)/5. When y = 0, the numerator must be zero, which gives -6 - 2x = 0 and leads again to x = -3. This confirms that the x-intercept is (-3, 0) and not any of the other candidate points.

Why Other Options Are Wrong:
Option b (3, 0) has the correct y coordinate but the opposite sign in x. Substituting x = 3, y = 0 into the line equation gives 2*3 + 5*0 = 6, which is not equal to -6. Options c and d, (0, 3) and (0, -3), are potential y-intercepts, not x-intercepts, because their x coordinate is zero. Option e (0, 6) also lies on the y-axis rather than the x-axis and does not satisfy the equation when substituted.

Common Pitfalls:
A common mistake is to confuse y-intercepts and x-intercepts and therefore set x = 0 instead of y = 0 when asked for the x-intercept. Another pitfall is mishandling the algebraic signs when solving for x, leading to 3 instead of -3. Carefully identifying which coordinate must be zero for each type of intercept helps avoid such errors.

Final Answer:
The line 2x + 5y = -6 cuts the x-axis at the point (-3, 0).