A line in the xy-plane has a slope of -{2/3} and passes through the point \((3, 5)\). Which of the...

1. TRANSLATE the problem information

• Given information:
  • Slope: \(\mathrm{m = -\frac{2}{3}}\)
  • Point on the line: \(\mathrm{(3, 5)}\)
  • Need: Equation in standard form (like the answer choices)

2. INFER the best approach

• Since we have a slope and a point, point-slope form is the most direct starting method
• Strategy: Use point-slope form, then convert to standard form to match the answer choices

3. SIMPLIFY using point-slope form

• Apply the formula: \(\mathrm{y - y_1 = m(x - x_1)}\)
• Substitute: \(\mathrm{y - 5 = -\frac{2}{3}(x - 3)}\)

4. SIMPLIFY the equation step by step

• Distribute: \(\mathrm{y - 5 = -\frac{2}{3}x + 2}\)
• Add 5 to both sides: \(\mathrm{y = -\frac{2}{3}x + 7}\)

5. SIMPLIFY to convert to standard form

• Multiply everything by 3 to eliminate fractions: \(\mathrm{3y = -2x + 21}\)
• Rearrange to standard form: \(\mathrm{2x + 3y = 21}\)

6. Verify the answer

• Check with the given point \(\mathrm{(3, 5)}\): \(\mathrm{2(3) + 3(5) = 6 + 15 = 21}\) ✓

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make sign errors when distributing or rearranging terms. For example, when converting \(\mathrm{3y = -2x + 21}\) to standard form, they might write \(\mathrm{2x - 3y = 21}\) instead of \(\mathrm{2x + 3y = 21}\), forgetting to flip the sign when moving the -2x term.

This may lead them to select Choice A (2x - 3y = -9) after additional arithmetic errors.

Second Most Common Error:

Conceptual confusion about slope: Students might confuse the slope \(\mathrm{-\frac{2}{3}}\) with \(\mathrm{-\frac{3}{2}}\), thinking the slope gives them the coefficients directly. They might write an equation like \(\mathrm{3x + 2y = [something]}\), misunderstanding how slope relates to the standard form coefficients.

This may lead them to select Choice D (3x + 2y = 19).

The Bottom Line:

This problem requires careful algebraic manipulation with fractions and negative numbers. The conversion from point-slope to standard form involves multiple steps where sign errors and arithmetic mistakes can easily occur, making systematic checking essential.