A line in the xy-plane has an x-intercept of 12 and a slope of 2/3. Which equation represents this line?y...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x\text{-intercept} = 12}\)
  • \(\mathrm{slope = \frac{2}{3}}\)
• What this tells us: The x-intercept of 12 means the line passes through point \(\mathrm{(12, 0)}\)

2. INFER the approach

• We need to find the equation in slope-intercept form: \(\mathrm{y = mx + b}\)
• We already know \(\mathrm{m = \frac{2}{3}}\)
• We can use the x-intercept point \(\mathrm{(12, 0)}\) to find b

3. SIMPLIFY to find the y-intercept

• Substitute the known values into \(\mathrm{y = mx + b}\):
  • Point: \(\mathrm{(12, 0)}\) means \(\mathrm{x = 12, y = 0}\)
  • Slope: \(\mathrm{m = \frac{2}{3}}\)
• Solve: \(\mathrm{0 = \frac{2}{3}(12) + b}\)
• Calculate: \(\mathrm{0 = 8 + b}\)
• Therefore: \(\mathrm{b = -8}\)

4. Write the final equation

• \(\mathrm{y = \frac{2}{3}x + (-8)}\)
• \(\mathrm{y = \frac{2}{3}x - 8}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept and think the line passes through \(\mathrm{(0, 12)}\) instead of \(\mathrm{(12, 0)}\).

Using point \(\mathrm{(0, 12)}\): \(\mathrm{12 = \frac{2}{3}(0) + b}\), so \(\mathrm{b = 12}\)
This gives equation \(\mathrm{y = \frac{2}{3}x + 12}\), leading them to select Choice (D) (\(\mathrm{y = \frac{2}{3}x + 8}\)) as the closest match, or causes confusion since this exact form isn't listed.

Second Most Common Error:

Conceptual confusion about slope signs: Students might think that since we're finding a negative y-intercept, the slope should also be negative.

This misconception might lead them to select Choice (A) (\(\mathrm{y = -\frac{2}{3}x - 8}\)) or Choice (B) (\(\mathrm{y = -\frac{2}{3}x + 8}\)), thinking the negative slope "makes more sense."

The Bottom Line:

The key insight is recognizing that an x-intercept gives you a specific point on the line, not just a number to plug in somewhere. Once you have that point and the slope, finding the equation becomes straightforward algebra.