Line h is defined by 1/5x + 1/7y - 70 = 0. Line j is perpendicular to line h in...

1. TRANSLATE the problem information

• Given information:
  • Line h: \(\frac{1}{5}\mathrm{x} + \frac{1}{7}\mathrm{y} - 70 = 0\)
  • Line j is perpendicular to line h
  • Need to find: slope of line j

2. INFER the solution strategy

• To find the slope of line j, I first need the slope of line h
• Then I'll use the fact that perpendicular lines have slopes that are negative reciprocals

3. SIMPLIFY the equation of line h to find its slope

• Convert \(\frac{1}{5}\mathrm{x} + \frac{1}{7}\mathrm{y} - 70 = 0\) to slope-intercept form \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\)
• Add 70 to both sides: \(\frac{1}{5}\mathrm{x} + \frac{1}{7}\mathrm{y} = 70\)
• Subtract \(\frac{1}{5}\mathrm{x}\) from both sides: \(\frac{1}{7}\mathrm{y} = -\frac{1}{5}\mathrm{x} + 70\)
• Multiply both sides by 7: \(\mathrm{y} = -\frac{7}{5}\mathrm{x} + 490\)
• The slope of line h is \(-\frac{7}{5}\)

4. INFER the slope of the perpendicular line

• Since line j is perpendicular to line h, their slopes are negative reciprocals
• Negative reciprocal of \(-\frac{7}{5}\) is: \(-1 \div (-\frac{7}{5}) = \frac{5}{7}\)

Answer: D. \(\frac{5}{7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make algebraic errors when converting from general form to slope-intercept form, particularly when dealing with fractional coefficients.

Common mistakes include forgetting to distribute the 7 correctly when multiplying both sides, or making sign errors. If they incorrectly identify the slope of line h as \(\frac{7}{5}\) instead of \(-\frac{7}{5}\), they would then find the negative reciprocal as \(-\frac{5}{7}\), leading them to select Choice B (\(-\frac{5}{7}\)).

Second Most Common Error:

Conceptual confusion about perpendicular relationships: Students may find the slope of line h correctly as \(-\frac{7}{5}\), but then find just the reciprocal (\(\frac{7}{5}\)) instead of the negative reciprocal.

This happens when they remember that perpendicular slopes are related by reciprocals but forget the "negative" part of the relationship. This leads them to select Choice C (\(\frac{7}{5}\)).

The Bottom Line:

This problem requires careful algebraic manipulation with fractions and a solid understanding of the perpendicular lines relationship. Success depends on systematic equation solving and applying the correct perpendicular slope formula.