Question:QUESTION STEM:In the xy-plane, line L is defined by the equation 3x - 4y = 8.Line m is parallel to...

1. TRANSLATE the problem information

• Given information:
  • Line L: \(\mathrm{3x - 4y = 8}\)
  • Line m is parallel to line L
  • Need to find: slope of line m

• What this tells us: Since parallel lines have the same slope, if we find the slope of line L, we'll have the slope of line m.

2. SIMPLIFY to find the slope of line L

• Convert the equation \(\mathrm{3x - 4y = 8}\) to slope-intercept form (\(\mathrm{y = mx + b}\)):

Starting with: \(\mathrm{3x - 4y = 8}\)

Subtract 3x from both sides: \(\mathrm{-4y = -3x + 8}\)

Divide everything by -4: \(\mathrm{y = \frac{3}{4}x - 2}\)

• The slope is the coefficient of x, which is \(\mathrm{\frac{3}{4}}\).

3. Apply the parallel lines property

• Since line m is parallel to line L, and line L has slope \(\mathrm{\frac{3}{4}}\), line m also has slope \(\mathrm{\frac{3}{4}}\).

Answer: \(\mathrm{\frac{3}{4}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when converting to slope-intercept form, particularly when dividing by -4.

Common mistake: \(\mathrm{-4y = -3x + 8}\) → \(\mathrm{y = \frac{3}{4}x + 2}\) (forgetting that \(\mathrm{+8 \div (-4) = -2}\), not +2)

This leads to identifying the wrong slope and providing an incorrect answer.

Second Most Common Error:

Missing conceptual knowledge: Students don't recall that parallel lines have equal slopes.

They might correctly find that line L has slope \(\mathrm{\frac{3}{4}}\), but then think they need additional information or calculations to find the slope of line m. This causes them to get stuck and abandon the systematic solution.

The Bottom Line:

This problem tests both algebraic manipulation skills and understanding of parallel line properties. The key insight is recognizing that finding one slope gives you both slopes when dealing with parallel lines.