In coordinate geometry, a straight line cuts the x-axis at the point (3, 0) and the y-axis at the point (0, 6). What is the equation of this line in Cartesian form?

Introduction / Context:
This question tests basic knowledge of straight line equations in coordinate geometry. A line is specified by its intercepts on the x axis and y axis, and you are required to find its algebraic equation. Recognizing how to compute the slope from two points and then forming the equation of a line is an essential skill in analytic geometry.

Given Data / Assumptions:
- The line crosses the x axis at point (3, 0).
- The same line crosses the y axis at point (0, 6).
- We assume a standard Cartesian coordinate system.
- We need the equation of the line in the form y = mx + c or an equivalent linear form.

Concept / Approach:
A straight line can be determined uniquely by any two distinct points on it. Given two points (x1, y1) and (x2, y2), the slope m of the line is (y2 - y1) / (x2 - x1). Once the slope is known, we can write the equation in slope intercept form y = mx + c by substituting one of the points to find the intercept c. Alternatively, we can use the intercept form x/a + y/b = 1 when the intercepts a and b are known.

Step-by-Step Solution:
Step 1: The two given points are A(3, 0) on the x axis and B(0, 6) on the y axis.Step 2: Compute the slope m of line AB.Step 3: m = (y2 - y1) / (x2 - x1) = (6 - 0) / (0 - 3) = 6 / (-3) = -2.Step 4: Use the slope intercept form y = mx + c. We know m = -2.Step 5: Substitute point B(0, 6) to find c: 6 = -2 * 0 + c, so c = 6.Step 6: Therefore, the equation of the line is y = -2x + 6.Step 7: Check with point A(3, 0): y = -2 * 3 + 6 = -6 + 6 = 0, so the point lies on the line, confirming the equation.

Verification / Alternative check:
We can also use the intercept form of a line: x/a + y/b = 1, where a and b are x and y intercepts respectively. Here, a = 3 and b = 6, so x/3 + y/6 = 1. Multiplying by 6 gives 2x + y = 6, which can be rearranged to y = -2x + 6, matching the slope intercept form derived earlier. Since both approaches agree, the equation is confirmed.

Why Other Options Are Wrong:
The option y = 2x - 6 has a positive slope and passes through (3, 0) but would intersect the y axis at (0, -6), not (0, 6). The equation y = 2x + 6 has a positive slope and does not pass through (3, 0). The equation y = -2x - 6 has the correct slope magnitude but gives a y intercept of -6 instead of 6. The equation x + 2y = 6 rearranges to y = (6 - x) / 2, which has slope -0.5, not -2, and does not pass through the point (3, 0).

Common Pitfalls:
Some students invert the difference in x and y when calculating slope, leading to the wrong sign. Others forget that the x intercept has y coordinate zero and the y intercept has x coordinate zero. Always write down both points clearly, compute the slope carefully and verify that your final equation satisfies both given points.

Final Answer:
The equation of the line is y = -2x + 6.