The equation 3x + 12 = 6y represents a line in the xy-plane. What is the x-intercept of the graph...

1. INFER what x-intercept means

• The x-intercept is the point where the line crosses the x-axis
• At any point on the x-axis, the y-coordinate is 0
• Therefore, to find the x-intercept, set y = 0 in the equation

2. TRANSLATE the problem into algebra

• Given equation: \(3\mathrm{x} + 12 = 6\mathrm{y}\)
• Set \(\mathrm{y} = 0\): \(3\mathrm{x} + 12 = 6(0)\)
• This simplifies to: \(3\mathrm{x} + 12 = 0\)

3. SIMPLIFY to solve for x

• \(3\mathrm{x} + 12 = 0\)
• Subtract 12 from both sides: \(3\mathrm{x} = -12\)
• Divide by 3: \(\mathrm{x} = -4\)

4. Express the answer as a coordinate pair

• The x-intercept occurs at \(\mathrm{x} = -4, \mathrm{y} = 0\)
• Written as coordinates: \((-4, 0)\)

Answer: A. (-4, 0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept and set \(\mathrm{x} = 0\) instead of \(\mathrm{y} = 0\).

When they set \(\mathrm{x} = 0\) in \(3\mathrm{x} + 12 = 6\mathrm{y}\):

• \(3(0) + 12 = 6\mathrm{y}\)
• \(12 = 6\mathrm{y}\)
• \(\mathrm{y} = 2\)

This gives them the point \((0, 2)\), leading them to select Choice B ((0, 2)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make sign errors during algebraic manipulation.

They might incorrectly solve \(3\mathrm{x} + 12 = 0\) as:

• \(3\mathrm{x} = 12\) (forgetting the negative)
• \(\mathrm{x} = 4\)

This leads them to select Choice C ((4, 0)).

The Bottom Line:

Success hinges on understanding that "x-intercept" means "where y equals zero," not "where x equals zero." The algebraic work is straightforward once this key insight is clear.