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 the form y = mx + c?

Introduction / Context:
This problem tests understanding of the equation of a straight line given its intercepts on the coordinate axes. The line is specified by two points, one on the x-axis and one on the y-axis, and we are required to determine its equation, ideally in slope intercept form y = m x + c.

Given Data / Assumptions:

• The line passes through point A(3, 0) on the x-axis.
• The line also passes through point B(0, 6) on the y-axis.
• We assume a standard Cartesian coordinate system.
• We need to find the equation of the line in the form y = m x + c.

Concept / Approach:
The equation of a line through two given points can be found using the slope formula. First, we calculate the slope m = (y2 - y1) / (x2 - x1). Then we substitute one of the points into the equation y = m x + c to solve for the y intercept c. Finally, we write the equation in the required slope intercept form and compare with the options.

Step-by-Step Solution:
Let A(3, 0) and B(0, 6) be the two points. Compute the slope m of the line: m = (6 - 0) / (0 - 3). m = 6 / -3 = -2. So the line has the form y = -2x + c. Use point B(0, 6) to find c: Substitute x = 0, y = 6 into y = -2x + c. 6 = -2 * 0 + c, so c = 6. Thus, the equation of the line is y = -2x + 6.
Verification / Alternative check:
Check with point A(3, 0): y = -2 * 3 + 6 = -6 + 6 = 0, which matches the given point. Check with point B(0, 6): y = -2 * 0 + 6 = 6, which again matches. Therefore, y = -2x + 6 is consistent with both points.
Why Other Options Are Wrong:
Option B (y = 2x - 6): Has positive slope. Through x = 3, y = 0 would give 0 = 6 - 6 = 0, but for x = 0, y = -6, not 6. Option C (y = 2x + 6): Intercepts and slope are incorrect; it does not pass through (3, 0). Option D (y = -2x - 6): Gives y intercept -6 instead of 6, so it fails to pass through (0, 6). Option A (y = -2x + 6): Correct equation, passes through both given points.
Common Pitfalls:
A frequent error is to swap x and y changes when computing the slope, changing its sign. Some students also mix up the intercepts and write equations like x = 2y + 6, which represents a different line. Another mistake is to assume c equals the x intercept, whereas in y = m x + c, c is the y intercept.
Final Answer:
The correct equation of the line is y = -2x + 6.