Line k in the xy-plane has a slope of -{2} and passes through the point \(\mathrm{(3, 1)}\). Which of the...

1. TRANSLATE the problem information

• Given information:
  • Slope of line k: \(\mathrm{m = -2}\)
  • Line k passes through point \(\mathrm{(3, 1)}\)
• Find: equation of line k

2. INFER the approach needed

• Since we have the slope, use slope-intercept form: \(\mathrm{y = mx + b}\)
• We know \(\mathrm{m = -2}\), so our equation is \(\mathrm{y = -2x + b}\)
• We need to find the y-intercept b using the given point

3. INFER the substitution strategy

• Since point \(\mathrm{(3, 1)}\) lies on the line, it must satisfy the equation
• Substitute \(\mathrm{x = 3}\) and \(\mathrm{y = 1}\) into \(\mathrm{y = -2x + b}\)

4. SIMPLIFY to find the y-intercept

• Start with: \(\mathrm{1 = -2(3) + b}\)
• Calculate: \(\mathrm{1 = -6 + b}\)
• Add 6 to both sides: \(\mathrm{1 + 6 = b}\)
• Therefore: \(\mathrm{b = 7}\)

5. Write the final equation

• \(\mathrm{y = -2x + 7}\)

Answer: C (\(\mathrm{y = -2x + 7}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the given point \(\mathrm{(3, 1)}\) with the y-intercept, thinking that since \(\mathrm{y = 1}\) when we have some point, the y-intercept must be 1.

They incorrectly assume \(\mathrm{b = 1}\) and write \(\mathrm{y = -2x + 1}\).

This leads them to select Choice B (\(\mathrm{y = -2x + 1}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when solving \(\mathrm{1 = -6 + b}\), such as getting \(\mathrm{b = -5}\) instead of \(\mathrm{b = 7}\).

For example:

\(\mathrm{1 = -6 + b}\)

\(\mathrm{1 - 6 = b}\) (wrong sign)

\(\mathrm{b = -5}\)

This may lead them to select Choice A (\(\mathrm{y = -2x - 5}\)).

The Bottom Line:

This problem tests whether students understand that any point on a line must satisfy the line's equation - not just special points like intercepts. The key insight is using the given point to find the unknown y-intercept.