A line in the xy-plane is defined by the equation 1/2y - 1/3x = 2x + 1. What is the...

1. INFER the solution strategy

• Given: \(\frac{1}{2}y - \frac{1}{3}x = 2x + 1\)
• Goal: Find the slope of this line
• Strategy needed: Convert to slope-intercept form \(y = mx + b\), where m is the slope

2. SIMPLIFY by moving x terms to one side

• Add \(\frac{1}{3}x\) to both sides to isolate the y-term:

\(\frac{1}{2}y - \frac{1}{3}x + \frac{1}{3}x = 2x + \frac{1}{3}x + 1\)

• This gives us: \(\frac{1}{2}y = 2x + \frac{1}{3}x + 1\)

3. SIMPLIFY by combining like terms

• Combine the x terms on the right side
• Convert 2x to thirds: \(2x = \frac{6x}{3}\)
• Now: \(\frac{1}{2}y = \frac{6}{3}x + \frac{1}{3}x + 1 = \frac{7}{3}x + 1\)

4. SIMPLIFY by solving for y

• Multiply both sides by 2 to eliminate the fraction coefficient:

\(y = 2 \times [\frac{7}{3}x + 1]\)

• Distribute: \(y = 2 \times \frac{7}{3}x + 2 \times 1\)
• Final form: \(y = \frac{14}{3}x + 2\)

5. INFER the slope from slope-intercept form

• The equation \(y = \frac{14}{3}x + 2\) is now in the form \(y = mx + b\)
• The slope \(m = \frac{14}{3}\)

Answer: \(\frac{14}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make fraction arithmetic errors when combining \(\frac{1}{3}x + 2x\).

Many students forget to find a common denominator or incorrectly calculate \(2x + \frac{1}{3}x\). For example, they might add \(2 + \frac{1}{3} = \frac{3}{3} = 1\), thinking the coefficient becomes 1x instead of \(\frac{7}{3}x\). This leads to \(y = 2x + 2\), giving a slope of 2, but since 2 isn't among the choices, this causes confusion and guessing.

Second Most Common Error:

Incomplete SIMPLIFY process: Students correctly combine terms to get \(\frac{1}{2}y = \frac{7}{3}x + 1\) but forget to multiply the constant term by 2.

They multiply only the x-term: \(y = \frac{14}{3}x + 1\) instead of \(y = \frac{14}{3}x + 2\). While this doesn't affect the slope calculation, it shows incomplete algebraic manipulation that could lead to errors in more complex problems.

The Bottom Line:

This problem tests students' ability to systematically manipulate linear equations with fractions. Success requires careful fraction arithmetic and methodical algebraic steps—rushing through the simplification process is where most errors occur.