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

1. TRANSLATE the problem information

• Given information:
  • Slope \(\mathrm{(m) = -2}\)
  • Point on line: \(\mathrm{(3, 1)}\)
• Need to find: The equation of line s in slope-intercept form \(\mathrm{(y = mx + b)}\)

2. INFER the best approach

• Since we have a slope and one specific point, point-slope form is the most direct method
• We'll use: \(\mathrm{y - y_1 = m(x - x_1)}\), then convert to slope-intercept form

3. TRANSLATE into point-slope form

• Substitute our values: \(\mathrm{y - 1 = -2(x - 3)}\)

4. SIMPLIFY to get slope-intercept form

• Distribute the -2: \(\mathrm{y - 1 = -2x + 6}\)
• Add 1 to both sides: \(\mathrm{y = -2x + 6 + 1}\)
• Combine like terms: \(\mathrm{y = -2x + 7}\)

5. Verify the solution

• Check: When \(\mathrm{x = 3}\), does \(\mathrm{y = 1}\)?
• \(\mathrm{y = -2(3) + 7 = -6 + 7 = 1}\) ✓

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when distributing the negative slope through the parentheses.

Students correctly set up \(\mathrm{y - 1 = -2(x - 3)}\) but then make the error: \(\mathrm{y - 1 = -2x - 6}\) (forgetting that \(\mathrm{-2 \times (-3) = +6}\), not -6). This leads to \(\mathrm{y = -2x - 6 + 1 = -2x - 5}\).

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

Second Most Common Error:

Poor INFER reasoning: Attempting to find the y-intercept directly without using point-slope form.

Students might try to substitute the point \(\mathrm{(3,1)}\) directly into \(\mathrm{y = -2x + b}\) to solve for b, but then make calculation errors like:

\(\mathrm{1 = -2(3) + b}\)
\(\mathrm{1 = -6 + b}\)
\(\mathrm{b = 7}\)

but then mistakenly write \(\mathrm{y = -2x + 1}\).

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

The Bottom Line:

This problem tests whether students can systematically apply the point-slope form and carefully execute the algebraic steps to convert to slope-intercept form. The key is maintaining accuracy with negative numbers throughout the process.