A line in the xy-plane is represented by the equation 4x/5 - 2y/3 = 7. What is the slope of...

1. INFER the solution strategy

• Given: \(\frac{4x}{5} - \frac{2y}{3} = 7\)
• Need: the slope of this line
• Strategy: Convert to slope-intercept form \(y = mx + b\) to identify slope \(m\)

2. SIMPLIFY to isolate the y-term

• Subtract \(\frac{4x}{5}\) from both sides:
  \(-\frac{2y}{3} = -\frac{4x}{5} + 7\)

3. SIMPLIFY to solve for y

• Multiply both sides by \(-\frac{3}{2}\) (the reciprocal of \(-\frac{2}{3}\)):
  \(y = \left(-\frac{3}{2}\right)\left(-\frac{4x}{5} + 7\right)\)
• Distribute the \(-\frac{3}{2}\):
  \(y = \left(-\frac{3}{2}\right)\left(-\frac{4x}{5}\right) + \left(-\frac{3}{2}\right)(7)\)
  \(y = \frac{12x}{10} - \frac{21}{2}\)

4. SIMPLIFY the coefficient of x

• Reduce the fraction: \(\frac{12}{10} = \frac{6}{5}\)
• The equation becomes: \(y = \frac{6}{5}x - \frac{21}{2}\)

Answer: \(\frac{6}{5}\) (or \(1.2\) as a decimal)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when multiplying by the negative fraction \(-\frac{3}{2}\), particularly when distributing to both terms. They might incorrectly get \(y = -\frac{6x}{5} + \frac{21}{2}\) instead of \(y = \frac{6x}{5} - \frac{21}{2}\), leading to a slope of \(-\frac{6}{5}\) instead of \(\frac{6}{5}\). This fundamental arithmetic error completely changes the sign of their final answer.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to rearrange into slope-intercept form and instead try to directly identify slope from the original equation \(\frac{4x}{5} - \frac{2y}{3} = 7\). They might incorrectly think the coefficient of x (which is \(\frac{4}{5}\)) is the slope, or get confused about what the slope actually represents in this form. This leads to confusion and random guessing among the answer choices.

The Bottom Line:

This problem tests whether students can systematically convert between forms of linear equations while maintaining accuracy with fraction arithmetic and negative signs - two areas where small mistakes lead to completely wrong final answers.